Confession: I badly wanted to leave your question without an answer (hence making it not a question?)!

But it was too good a question. Or two good questions: (1) could there be questions to which there are no answers? And (2) could there be questions which have answers which we will never know?

If you think that it makes sense to say that reality isn't always determinate, isn't always a particular way, then the answer to (1) would be Yes. There are some questions in mathematics, for instance, to which some (but not all) mathematicians believe there is no correct answer; the Continuum Hypothesis is a famous example. Some (but not all) physicists hold that some features of the physical world are indeterminate as well, for instance that there simply is no right answer to the question of what precisely a particle's position and velocity are at a particular moment. On the other hand, if you believe that if a question is fully meaningful, then the relevant reality it concerns must settle the matter one way or the other, then (1) would have to be answered No.

The question (2) also gives rise to heated debate. As you can see elsewhere on this site, there are philosophers who believe that the fact that our minds have the particular structure they do makes it inevitable that some truths about our world will remain forever beyond our grasp. On the other hand, there are philosophers who think that any question whose answer is in principle not knowable by us is really not a clear question. The great American philosopher W.V. Quine (1908-2002) puts it in his inimitable way:

Questions, let us remember, are in language. Language is learned by people, from people, only in relation, ultimately, to observable circumstances of utterance. The relation of language to observation is often very devious, but observation is finally all there is for language to be anchored to. If a question could in principle never be answered, then, one feels that language has gone wrong; language has parted its moorings, and the question has no meaning. On this philosophy, of course, our central question has a sweeping answer. The question was whether there are things man could never know. The question was whether there are questions — meaningful questions — that man could in principle never answer. On this philosophy, the answer to this question of questions is no. [From "The Limits of Knowledge" in his The Ways of Paradox and Other Essays (revised and enlarged edition).]