There are questions I don't like (i.e. the "does Johnny love me?" variety) but there aren't many questions that I'll refuse to answer. In fact, depending on how deeply I go into the divination, into the flow of wyrd, the words might just tumble out before I can even consider not answering. If i'm taken up as an oracle, that is a thousand times more the case.
There are two exceptions to this.
A. Sometimes the divination will show that the client should not be told a thing -- for whatever reason. Quite often, it's a matter of the client needing to gain the power or the character development that will come from sorting a thing out for him or herself; or perhaps knowing in advance will shift the wyrd too drastically in ways that will not benefit. Sometimes diviners will get what we tend to colloquially term 'gag orders.' We'll go to attempt to tell the client something, and the Gods or ancestors will simply not permit it. It can be rather amusing to watch a diviner try to speak a thing, only to actually choke on the words, or as a diviner to open my mouth to say one thing and have something totally unexpected tumble forth. It can be frustrating for client and diviner alike to be unable to answer a question, but when this happens -- be it a 'gag order' or something coming up in the reading that specifically indicates that the reading should stop or that a question cannot or won't be answered--there's usually a compelling reason.
B. The other occasion on which I will refuse to answer a question is a bit more nuanced. I won't precisely refuse, but if someone asks a poorly crafted question, I will often prompt the client to rephrase or will flat out tell him or her that I am not going to ask that, but will instead ask X (at which point I will rephrase the question, often explaining to the client why I chose to do so). For instance, I once had a client who came to me because she had been having some very intense experiences with one of her Deities. Toward the end of the divination, she asked me if her Deity had laid a taboo on her, forbidding her to eat meat. She said "Do I have to be Vegan?" I knew, just from the flow of the reading and the feel of the divination that if I threw the answer would be an absolute yes. Sometimes, as a diviner, you just know what's going to come up on the mat. I told her that I wasn't going to ask that. Instead, I was going to "do her a solid," and ask if she should limit her intake of red meat as a devotional act. The answer came up an unqualified yes and then we worked out a few more specifics. This is a fairly simple example but what I did there was keep my client from running head-first into what for her, would have been a difficult and unpleasant (and possibly unhealthy for this particular person) taboo. Had I asked her original question, and gotten an absolute yes, then she would have been obligated. Instead, because of the way the question was phrased, i was able to work out something much more livable. Whether I do this or not is completely at my discretion because it's the client's job to ask appropriate questions, but generally I will if the chance arises.
More often though, if there is an issue with a question, what will happen is that the divination will not answer the client's question. Other things are deemed more important, sometimes factors and consequences the client can't see, or sometimes a completely different issue all together, and that will come up rather than clearly what the client asked. That is vexing and sometimes switching to a binary system will allow a workaround (sometimes not).
I suppose there is one type of question that I won't answer, and that is a question so egregiously miasmic, hubristic, and polluted that it would be an offense even to bring it to the mat. I will also refuse to answer silly questions, like 'what should I have for dinner, chicken or fish? " (and yes, this is stupid, but I also know of a case where someone asked this -- not of me thankfully). Such questions are disrespectful on so many levels and frankly, stupid.
I also will flat out refuse to read for someone who is chasing the answer he or she wants, going to diviner after diviner with the exact same question. I may also refuse someone who comes to me three or four times with the same question, having taken none of the advice given in previous readings on that topic. What these have in common, by the way, is that they show no regard whatsoever for the sacredness of this practice.