Question: Suppose that I posted, as an answer, a fairly good hint to a question, but the OP still can't solve it on their own, and they ask me for further details. Would it be rude if I did not reply?

The only reason I'd choose to do that would be foreseeing that it would lead to me eventually giving out a detailed, complete answer, and I don't want to do that because it is the kind of questions where the details can be easily filled in if the student understands the underlying material well.

Answer: 
Nope. Nobody has the right to usurp your time. In fact, it is rude of someone to expect that you drop everything to answer every single follow-up question they may possibly have. You are here for your enjoyment, not out of a debt to anyone.
It depends on the circumstance. If you post one comment or answer (a hint, as you say) and the OP requests further clarification in the comments and you choose not to reply, I believe that is rude. Facilitating mathematical discussion is the reason we have a "commenting" system; if you're uninterested in discussing, don't answer in the first place. If you've given a hint and you believe further hints will lead to an answer, explicitly say so; you don't need to ignore. From a pedagogic standpoint (this is based on my personal experience tutoring), in most cases, it's possible to elaborate on hints without giving a full solution. 

On the other hand, if after several comments it's clear that the discussion is going nowhere and it's fruitless for you to comment any further, in this case, I feel, not replying would be appropriate, though it's best to explicitly say you will not comment any further.