Personally, I like to play devils advocate, and I'm curious about what happened on 9/11. It doesn't add up for me. So I have taken the "conspiracy theorist" side several times... I don't see people making arguments against the theorists data.
I see the same common fallacies, most often variations of appeal to authority. "Popular science debunked that" for example in response to an argument that Popular Science didn't address.
"If it were a cover up it would have gotten out" in response to people pointing "it getting out" is my favorite, of course. Sort of a "I don't believe it, therefore it can't be right". Not sure what fallacy that is.