The situations where you are asked 'do you have any questions?' are often the more stressful ones in life. Therefore it could be helpful to have some preparation of the types of questions you may wish to ask.
They say lawyers say 'never ask a question you don't already know the answer to'. This may be useful in court and less so in real life, but it does pay to think about what kind of response you are likely to receive before opening your mouth.
Someone once gave me the idea that questions can be categorised. Here is my attempt to do that. There are certainly others, which I may add another post about. Comments are also welcome.
1) Ask for background on speaker/issue discussed
1a) sub-category to the above is to ask clarify something
2) Ask about extensions to the problem
3) Questions that exalt the questioner
4) Hypothetical scenarios
5) Asking to obtain something
6) Goading questions
Explanations and examples.
1) Asking for background can help the listener put the comments in context and so understand them better. This kind of question can range from 'tell me about where you worked before' to 'what is your approach to people management?' to 'what has influenced your life philosophy?' or even 'have you ever...?'/'what is the most x you've ever...?'.
1a) This one's fairly simple. The listener hasn't understood something or found something unclear. The effect of asking this kind of question is often relief or derision from the rest of the audience and the speaker, depending on the level of knowledge of the same. However, don't let fear of derision put you off: If the speaker is worth their salt, they should feel (i) foolish for not explaining better first time (ii) glad for the chance to help people understand.
2) This is can be unimaginative. For example, 'you did this problem in 2D -- how do you think it would work in 3D?'. On the other hand, if you genuinely have an idea that could help or bring enlightenment, the speaker might be pleased to hear about it (example: couldn't you just use the xyz and theta variables from the abc dataset then you can calculate the one you need by taking the blah blah?). Or the speaker may get defensive and not appreciate the help, which could be seen more as interference/showing off (see 3).
3) Showing off is not cool, man.
4) This can reveal insight (see 1). For example, the classic 'how would you deal with a difficult customer?' types to the totally hypothetical 'what book would you take if you were to be stranded on a desert island?'/'what would you do if you won the lottery?' types. But more useful are the questions that reveal and provide insight into the speaker's way of thinking (see 1 again), so tailoring the imagined situation to the ideas you a trying to elicit. No point in asking a political candidate about books on desert islands, but plenty in asking about their actions in the event of an election victory.
5) This category is similar to 1a), in that it's merely fact-seeking. Not necessarily dull, though, for example if asking for a salary increase (for advice on which, see this wonderful site).
6) See journalists and also 3. Not cool. Reveals more about the ignorance of the questioner. Deliberately trying to make someone feel uncomfortable is un-zen. Respond with zen-ness and you will come across well (e.g. 'that's an interesting point, I'll have to give it some more thought', or 'I've never seen any evidence to support that, but I may need to do more research' is a fair response to many of this type).
Finally, remember the wise Chinese proverb: he who asks is a fool for five minutes; he who does not, is a fool forever.