Second draft (please see my other answer for context):
What about open problems?
It depends on what is meant by an open problem: here, we mean a problem which is well-known to be unsolved and very hard. Questions for which you simply do not know the answer are not the issue, as those are the norm on MathOverflow.
On the one hand, a question which simply states a famous open problem will often not be well received [[comment: this may sound too harsh...]]. If you mainly want to record some open problem, this site is not the appropriate place: you may do so at the Open Problem GardenOpen Problem Garden, for example.
On the other hand, questions that touch upon a well-known open problem are certainly welcome, as long as they follow the general guidelines for asking on MathOverflow [[comment: perhaps placing a link here to the guidelines would be a good idea]]. Nevertheless, when asking about an open problem, you should mention that it is known to be unsolved. In this case, please use the 'open-problem'open-problem tag, in addition to other appropriate tags.
Most importantly, please remember that MathOverflow is a question and answer site: the prototypical question is posed under the assumption that some other user will be able to answer it. This principle also applies to questions about open problems. For example, the community will usually welcome inquiries on partial progress, results in special cases, etc.
What if my question turns out to be a well-known open problem?
If you ask a question which is widely known to be unsolved, someone will typically point that out and provide references to the relevant literature. In most such cases, this is the best answer one can expect. At this time, you can add the 'open-problem'open-problem tag to your question. If you then have more detailed follow-up questions, feel free to ask them in new posts.