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.
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.
In the specific case you describe I second what J.M. said. In my opinion the most apt course of action is to state that you do not wish to elaborate.
This has several advantages:
OP knows there is no point in waiting for you to reply.
Others will know that you decided actively not to elaborate further and this might very well inform what they do. It is common for other user to step in and provide additional details. I do this frequently. I do this to avoid the wait until the original answerer comes back. I usually would not if I knew they actively do not want to give more information out of some pedagogical consideration.
Thus, leaving considerations of etiquette it is efficient to be explicit regarding your intentions.
It depends on context but, generally, I do feel it is a bit impolite not to acknowledge a request. Not to do what is requested is fair game, but at least you should say so.
Of course, such as there can be exceptions, if there were earlier interactions or the answer is explicit about it not being developed etc.
I also agree with hardmath that if you find yourself often in that situation you may want to look into the choice of the questions you answer. The point of the site actually is for the most part to give ready-to-use answers. This is a vague notion, but it still means that it is in principle not at all unreasonable that somebody asks for further details.
Certainly, some user behave in unreasonable ways, but one request for further details is pretty reasonable and should be treated as such and not as an attempt to "usurp [ones] time" etc.
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.
I believe that simply choosing not to reply is a bit rude. I would support leaving a comment such as "Please put more effort into this question. I would rather not post a full solution" instead of not replying.
I understand your point that sometimes you feel that the student is not giving a minimum effort, and it is true that you have no moral obligation to reply and elaborate. The best course of action however, in my eyes, as a student, is to reply by saying something along the lines of:
"what I have written is a direct basic application using the definition of this or that, revise them and I promise the hint should largely be sufficient".
Think about what you would want, if you were asking the question. (You should be able to figure out the answer from this fairly good hint.)
