Fantasy solution:

The bridge to Heavens is above the pit to Hell. Anyone pushed over just vanish into its bottomless abyss. Just rule out accumulation.

Biological solution:

Scavengers. Some scavenger nest near the duel area and clean the corpses once sun has set. They learned to be afraid of the guardians after centuries of getting their butts whooped, but are extremely hungry for freshly slain humans. They even eat bones and are numerous enough to only leave smears on the floor. Fallen heroes are quickly retrieved by the terrified villagers to be buried properly.

One advantage to this solution is that the scavenger population should regulate itself to adapt to the income of new sacrifices contestants.

how could they dispose of the many corpses?

Don't.

A gigant pile of rotting corpses floating on their own goo may be a nice deterrent against stupid adventurers trying their luck. If the sight doesn't scare them away, the smell will. They will get sick. Any adventurer who escapes will die from infection far away from the site and scare others even more.

And since the guardians are immortal, they won't be affected much. The smell might be bad but millenia of exposure may do for some resistance.

Let's do the math.

• Average human volume is 0.066m^3. Let's assume they're big people, at 0.1 cubic metres.


• Let's also assume no reduction in volume over time through mummification or decomposition: the pit's frozen, perhaps.


• Bingham mine is a pit about 940m deep and 4km across, and the highest cliff in the world is 1340m high. So we know that a 1km cliff is feasible.


• Assume that the cliff has no karst already at the bottom, and none of it wears down over the millennia. Which is a cheat, but meh.


• To stack up against a cliff, you need a half-cone with an angle of whatever angle is whatever angle of repose your material has.


• I'm gonna just go ahead and assume 45 degrees AoR for these frozen bodies. Most stuff is lower, I know nothing that's steeper, and steeper means fewer bodies, so 45 is likely our worst case.


• Volume of a cone is 4 pi r^2 height/3.


• At 45 degrees, height = radius.


• So 4/3 pi cubic km: roughly 4,188,790,200


• You need 40 billion people to make this pile.


• There are about 7.7 billion people in the world.


• About 110 billion people have ever lived.

Assuming that everyone who ever died was slain by these guards, we would have had to be throwing everyone from the last thousand years at them in order for the pile to have stacked up against the cliff.

So basically, if it's just a few nearby towns, which are having their populations artificially lowered by sending people here to die, then there's no earthly way that they'll ever pile up high enough to be a concern for the guards.

And if you take away the constraints against decomposition and desiccation, the numbers become many times more crazy.

Option 2: the guards can just trample them down as they patrol up and down the path to the gate. This will build up a berm with a walkway on top, and with frozen bodies piled up a little (less than one person-height) either side, of the people who had fallen to the wayside.

Bodies falling in the road would become trampled, raising the roadway with their 0.1m^3 of volume.

The berm would slowly rise from the plain, with sides angled at about 45 degrees, from bodies sliding down from the piles at the roadside. Cross section would be like:

   /~___/~

  /#########

 /###########

/#############

I think the number of bodies needed to build up a section of patrolway would be:

(roadway width + 2 * roadside pile width + height) * height * length * bodiesPerCubicMetre.

So for a 1km long roadway, 10m wide, with 2m piles either side, piled up 100m, that'd be 114M bodies. Not counting the bodies needed for the ramp at beginning and end.

Pile it up to 1km and it's 10Bn, though there'd be definite stability problems before that point, if only because the path width is now greater than the length. Plus, the gate to heaven would probably be buried in bodies by this point.

Option 3: Woodchipper.

So we move it from the frozen tundra to somewhere a little warmer, and make the guards just insanely violent. Or give them access to a wood-chipper, same difference.

There's just not enough of their victims left at the end of the fight to bother with, especially when what's left of them has been marched upon in metal-soled boots a few times.

They are a wet paste which seeps down into the earth, oozes away into the water table, washes away in rain. Yes, eventually all the bonemeal will raise the ground level a little, but not significantly, and the angle of repose for rained-on corpse-slime is unlikely to be much more than about a degree, so there won't be a steep hill anywhere.

Crows and rats will likely sup happily on the ooze, but will also shit it out, so won't affect volume much.

Streams might carry some away, but that's just natural erosion and applies to any land over thousands of years.

The bigger problem will be that corpse-ooze will start eventually seeping through the gateway to heaven. If Heaven doesn't want to start stinking on a hot day, there better be a very long stairway to heaven inside the gate.

The path to heaven is at the top of a mountain, at the end of a long treacherous trail that leads over several smaller mountains. The region is isolated (though not terribly far from settlements), mostly above the treeline, and filled with endless ravines.

Coiling Dragon Cliff Walkway on Hunan’s Tianmen Mountain, China

Places like this exist in the real world. The mountains are steep enough that it's easy to toss bodies off without having them land close enough to make a mess or a smell. And tall enough that millennia of body tossing will never bring the piles high enough to see from the top.

Because the guardians are hard to reach, not many will try. Even a million years of dead bodies won't fill up the valleys. They will mostly decompose and compress anyway, with some animal scavenging, so they won't take up as much space as they would if the bodies were all fresh.

The guardians won't have to do much to dispose of the bodies. Just a push and a roll and off they go. Once in a while, they might have to use a long pole to remove bits stuck to the side of the mountain, but the cliff sides are fairly smooth there and it's only rarely a problem.

The corpses stay.

After a short time on the ground the corpse of the defeated gets back up and walks a short distance to a dusty field. It then sits crosslegged and watches. It will watch until the end of time.

If a given corpse is damaged such that after defeat it cannot make this walk, others from the field will rise and help it to its place.

Many are seated in this field. They failed to gain entrance to heaven, and so they stay where they failed.

There is no issue really.

The religious authority arranging the event equips the people and escorts them to the site. With criminals armed guards are used. With volunteers priests and musicians.

The guards are then challenged by the victim. The victim is killed and his escort gathers up the corpse and the expensive equipment and carries it back.

I mean why would they leave expensive equipment there when they can just recycle it? That stuff is expensive you know. It would never be left there to rust. Even if it gets broken the metal it is made of would be too expensive to lose.

And what kind of religious ritual does not include proper burial rites for the sacrifice? Religious rituals work much better if they include parts that the common people can witness around the sacred parts. One ritual prepare for the challenge, another when they leave the town, third when their body is carried back to town, and one last one when they are buried.

I think the part you missed is that when you made this a ritualized practice, it became a ritualized practice and that includes taking care of sustainability as part of that practice.

Make the ritual be more like a formalized duel:

• 
  Most fights won't be to the death. It's an honor to hold a fight against the guardians. Only a small percentage will want to fight to death. Most will yield after first wound.


• Have the duelist be assisted by a second whose job is to testify of the actions, the outcome and if the outcome was death, take care of the corpse.


• Have the town that is nearest to the bridge enforce the rules. Anyone crazy enough to not follow them will be killed before reaching the bridge.

EDIT:

OK. Given the comment that fights indeed will be to death, it makes not much sense to send your best warriors but I still thing the duel "second" can make sense if it's some kind of ritual.

I would also add that the nearby towns would be quite disgusted by the increasing rotting pile of corpses and the increased scavenger activities and may want to voluntarily dispose of the corpses themselves. If indeed the warriors usually are equipped with the best armor and weapons they can afford, recovering these artifacts can be a way to offset the cost of cremating the corpses.

I also can imagine the town sprouting an industry around the "bridge to heaven", with people who will insure your body will arrive intact to your kin -for a fee-, used equipment sellers "It's been only used once! Almost without scratches!", and lots of unsavory guys of the same type...

The fight is very close to (or above) a large river. The corpses are tossed in and dragged away by the current. There the local fish and wildlife living on the Banks can deal with it the carrion way.

Other options would likely need more special circumstances. Carrion eaters and scavengers are likely seasonal, and would not fulfill their body quota during certain times.

Speaking of seasons. Another option could be above the eternal snow line. Mount everest for example has dead bodies on it of people that froze to death, and the bodies are too hard to remove and will remain there. The cold is do harsh they dont decay and many of the bodies are literally used as markers now. Use this for a more grisly tale: the killers stack the bodies as walls. Anyone who tries to defeat them has to walk through a wall of frozen bodies. Should you try to reach the killers while one is dragging a body (maybe he has a small cart?), you'll have to pass them somewhere inbetween those walls and fight them anyway.

The guards may be immortal, but they still need to eat. Or perhaps they don't need to eat, but they like to. The towns may call the people they send "heroes", but the guards call them "dinner". Most of the problem of corpse disposal is what to do with the rotting flesh - and the answer is "eat it before it rots".

So the problem then isn't what to do with the corpses as such - the problem is what to do with the bones after the guards have eaten. Ground bone has a mineral content not dissimilar to limestone so would make a good mortar (mixed with the blood of recent victims), and the larger bones can become building material.

Over time, the guards build themselves an ossuary tower from the bones of the fallen. Perhaps the guards still feel a duty to memorialise the valiant fallen and ensure contenders do not take the attempt lightly - so the exterior of their tower might be
lined with the skulls of victims, filled with bone mortar and small bone gravel so that they are structurally sound for building.

Manipulate biological processes.

Since the guards clearly have the ability to slow the speed of biological processes (that's why they don't age), they they can utilize the reverse ability and speed up biological processes for the corpses. If you make a corpse age a thousand years overnight there isn't going to be much left of it. (Yes there have been corpses with identifiable parts after hundreds of years, but that's usually in closed containers).

If I was in charge of the kiler guards, I would equip them with some type of body disposing weaponry. If you are killed by their weapons, you dissolve into a fog which drifts away along the path.
Or, as cool as these super-killers obviously are, perhaps they do not even need magic weaponry, perhaps their unique killing skills are so out-of-this world that you don't just get killed by them, it's not a simple "heart stops beating, soul leaves body"-thing, it's a complete disintegration of the entire being. "I saw the sword hit the bearded man, there was a sound - like a scream abruptly stopped - and then the man was simply gone! You could still see his footsteps on the path, but other than that it was as if he had never existed..."

Pay to play. Any challenger will have to clear the killing field first. This may be bound into a ritual, it may be tradition, or just something the guards spring on the champions as they walk in.

If the guardians are fighting on a bridge above a river they could simply roll the bodies off the bridge and into the river to have the current take the bodies away.

Maybe the introduction of another critter that carries off the remains at eats them. It certainly would be a deterrent as you watch your former fallen comrades being devoured, and you still have a few miles to walk before you get to the guards. The critters of course, just watch you go by because why bother with the effort of killing, when you will be dead soon enough, and more to the point, they are already very very well fed. So of course, the landscape is now littered with piles of bones and maybe the young of the carrion eaters fight over scraps for play.

Hey Bob pass me that leg.

These guardians might not be inactive when no one is around. After the poor sap(s) have been dispatched the guardians go around eating every part of the sap(s).

Alternative:

Intention matters

Only those with the intention of going up the path to what is believed to be heaven is attacked. If you are there to pick up the body of your brother/sister than the guardians would not even be aware of your presence. Same would be true for farming / building / ...

You could have the gate/path in a massive city. As long as people don't care about crossing the gate / threshold they are fine to go about what ever they want.

If not in a city: cannibals / scavengers would live in the area to feed off of this and bye bye bodies.

All you need is gravity.

Make the path narrow and at least one side of it very steep and free of obstacles. Gravity will do most and indeed often all of the work.

The longer the slope/higher the cliff, the further any decay or mess gets removed from the battleground. What's more, you've a wide variety of ecosystem options to manage the corpse's destination long-term.

That open-air gravesite can even contribute interesting plot elements through that location's ability to: build the legend; nourish unique vegetation or wildlife; harbor its own unique dangers; present unique or insurmountable accessibility challenges; fuel the trade of used equipment; be the prize in local gang wars; etc.

As for the rate and volume of corpses accumulated, I don't see that being a problem unless this location is remarkably small for its supposed importance. I would expect greater logistical problems managing the path's summit or beginning, as it will be host to multiple sizeable delegations daily, which (in order to sustain this rate) must necessarily be often from considerable distance. This will mean: high volumes of human traffic needing amenities both at the destination and en route; conflicting factions bumping into each other (and at least sometimes outside areas considered sacred and strictly peaceful); uncomfortable scheduling conflicts with "event planning" (particularly between different types of hopefuls); etc.

Nature has a way of turning challenges into opportunities. If your warriors provide a festering pile of corpses, nature will provide massive swarms of carrion beetles. The nice thing about using life to clean up death is that the cleaning force will scale to the available food supply. Whenever bravery/foolishness runs rampant, the food supply and resulting swarm grow. When cowardice/wisdom rises, most of the beetles starve away, leaving a few stragglers behind to wait for the next feast.

A Modest Proposal

The guardians not only guard the Way, they receive immortality in return for their vigilance. Specifically, so long as they are on the job, when they kill a challenger, they are able to replenish their youth and strength from the corpses of the fallen, by feasting on their remains (either traditionally, by mouth; or metaphysically, by absorbing them, quickening, body, and all, as they fall). In so doing, the strength of the fallen gets added to their strength, as do the days remaining until they would have died, had they not come to do battle.

This has a number of effects.

First, a death row convict is not as filling a meal as a warrior seeking glory; had they not opted to challenge the guardians, their days would have been numbered. Of course, a warrior seeking glory might have had a short lifespan in other battles as well.

Second, if the surrounding communities were to decide that they simply must get to heaven, they could come to an agreement to stop all challenges for a period of time; by having all challengers instead train in anticipation of a future day when they will make an all-out attack, they will increase their own cumulative strength while starving out the guardians. Perhaps they will succeed at that point or not; but they would certainly stand a better chance than continuing attempts as they always have.

Third, what if there used to be three guardians? Or twenty? Did the remaining two perhaps eat their own comrades who fell either to challengers -- or to their own comrades after giving into weakness and trying to travel the path themselves? What would either option mean for the future of the guardians?