I'm finding the question a little bit vague. But as I understand it, the general principle, which actually applies to all sites in the StackExchange network, is to have crisp and to-the-point question-answer interchanges based on facts (verifiable if possible). In fact the software is designed to avoid long, rambling discussions or arguments.
How this plays out in practice is site-dependent. We occasionally have for example questions that ask for advice (on co-authoring papers, say, or textbooks for a certain course). Some of those remain open, some are closed. It's hard to say exactly where the dividing line is, but one factor may be asking questions based on personal experiences (hence factual) as versus others which are primarily opinion-based.
Echoing the question of მამუკა ჯიბლაძე (Mamuka Jibladze), it might be easier to address this question with actual examples in mind. I actually disagree with the assumption that MathOverflow "doesn't like" opinions, but how we deal with them... it depends.
(And now I see Martin gave an example in response. I think the community response is consistent with what I explained above. It seems a hard question to answer [define "fruitful" and "how often" -- many people here think about mathematics all day long].)
