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In The Simpsons opening Homer handles a rod of uranium, that he gets rid of while commuting to home afterward. However, as he works as a safety inspector, it doesn't seem to be part of his work responsibility to deal with uranium directly. Is there any explanation why he does that?

Homer's first job was actually not a Nuclear Safety Inspector. He was a "nuclear technician" or "technical supervisor" (No one really knows. Not even him). In S01E03, he is fired from this job. He eventually becomes a "safety crusader" after witnessing all the safety violations the plant commits. Mr. Burns re-hires him as a safety inspector to shut him up.

That part of the opening sequence has been unchanged for the entire span of the show. What you're actually seeing is Homer at his original "Nuclear Technician" job before he became a safety inspector.

DeeV answered it quite well but one more update it's not even rod of uranium but a carbon rod. And to be accurate Inanimate carbon rod.

There were two such rods prominent in the show as detailed in linked wikia. And it shows he is less valuable to the plant than an inanimate object and also during NASA mission history repeat itself.

And he did use to work in Springfield Nuclear Power Plant, so having a carbon rod is not so strange especially when you envy it so much.