1. The whole birth thing has become a conspiracy theory whose most vocal propagators seem to be bona fide crazy (check out the video here). In reality, I imagine that the only reason this issue has legs is because some people vigorously don't like him as president and this has become a legitimate outlet for them. In fact, I bet there's a large intersection between the group of people who strongly don't like Obama and the birth certificate people.
What could be greater politically if your loudest opponents are projecting a crazy which repels most middle-ground people? In other words, I think the core of his opposition is isolating itself from typical people which would effectively contain it. He'd be a fool to stop it.
Not only that, it's also a convenient distraction for him for the public debate to be dealing with this low-level stuff instead of the big picture.
2. Along the same lines, as with other conspiracy theories, like 9-11, or the moon landings, the Loch Ness monster, etc, there's plenty of documentation one way or the other and people still take unreasonable positions. I doubt they'd stop if even more stuff were produced. They'd brush it off as being forged and continue, or maybe move on to something else about him.
Relevant: FAQ from the BBC
It's ridiculous and that's all I'm going to say about that.