I have been thinking about this for a few days (mostly because I would headhunt Refute This! questions because I love debating; even small details). However, I agree with the community that this site should avoid the "Oh, yeah, well, this and this ..." attitude. But something feels missing.
The "Refute This" type question holds a certain value, I think, that we would not get if we disallowed any semblance of it. That value is deep exploration into the thought process of specific viewpoints. So I thought of the "Why do they believe this?" question. Do we not all hear "these people believe x" and sometimes wonder why, saying "how did they come to that conclusion?" The weaker version of this is "is it true that these people believe x" which the community clearly allows, but it is such a boring question and asks for very little. Answer: yes. Whoopee! That was exciting! Not.
Does the community think we should allow "Why do they believe this?" questions? I think if we do then those types of questions need to be closely watched to prevent them from falling into debate in the comments. Push that into a chat room. Perhaps they should automatically be protected.
Justification Questions does point out a legitimate concern with Why do they believe this questions: That they might discourage users from returning, feeling like they are constantly under attack.
To sum a point that Jon explains well: Refute This is challenging by its very nature and is generally unproductive. Why do they believe this is truly inquisitive and encourages a positive response.