Could the phrase "splitting hairs" (i.e. focusing on details that are
not important) be considered a logical fallacy? If so, which one?

I think it comes pretty close to the definition of a red herring:
http://www.infidels.org/news/atheism/logic.html#herring

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Request for Question Clarification by denco-ga on 07 Jun 2005 18:49 PDT

First, yes, I think that "splitting hairs" would be considered a logical
fallacy, but I think it is more an example of the following:

Argumentum ad nauseam
http://www.infidels.org/news/atheism/logic.html#nauseam

By "splitting hairs," someone is just repeating the same thing over and
over again, but stating it somewhat differently or in a different detail
or manner each time.

A "red herring" is more of a misdirection than repetition.

Does this make sense?

Looking Forward, denco-ga - Google Answers Researcher

"Splitting hairs" does not necessarily involve repetition. For
example, if a professor fails a student's exam for having a spelling
mistake, he would certainly be splitting hairs; however, he would not
have repeated anything at all.

Depending on how the phrase is used, it may or may not be a logical
fallacy. If it is a fallacy, it is likely the fallacy of
insignificance.
http://www.datanation.com/fallacies/insig.htm