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This is the sentence:

Entrepreneurial leadership is the leader’s ability to influence others, to nurture the culture, to manage resources, and to develop a competitive behavior to seek opportunities and advantages (Ireland et al., 2003).

In my language, we don't repeat the equivalent of "to" multiple times and it's correct to use it just once (before the first verb).

So I was wondering, would the following sentence be grammatically correct:

Entrepreneurial leadership is the leader’s ability to influence others, nurture the culture, manage resources, and develop a competitive behavior to seek opportunities and advantages (Ireland et al., 2003).

Or is it necessary to use "to" before each verb?

In your sentence, all the additional "to" seem redundant. But there are some instances where adding additional "to" may be necessary.
For example, in legal documents, when we are empowering someone to do certain acts, we use "to" after every comma to indicate that now a new power is being listed after the comma. Same goes for setting out duties. This is also found in laws.

Now, in this context, you would use additional "to" if you are not defining the entrepreneurial leadership but writing an instruction or expectation for someone to be hired as an entrepreneurial leader.

The example you gave, of a sentence with fewer instances of the word “to” seems perfectly fine to me, as a native English speaker and grammar hobbyist.

I'm inclined to object to using "to" so many times in a single sentence because of general admonishments against redundancy in Strunk and White's "Elements of Style" among others.

Here's what Strunk and White wrote that seemed relevant in this case:

17. Omit needless words. Vigorous writing is concise. A sentence should contain no unnecessary words, a paragraph no unnecessary sentences, for the same reason that a drawing should have no unnecessary lines and a machine no unnecessary parts. This requires not that the writer make all his sentences short, or that he avoid all detail and treat his subjects only in outline, but that every word tell. Many expressions in common use violate this principle.

However, I understand that this does not address the grammatical validity of the shorter sentence.