I think--as Bill Clinton showed during Lewinskygate--that prosecution of a sitting President becomes a political act. One in which the court of public opinion ultimately decides whether or not that President is guilty. It has become very difficult to "shame" a President out of office, a la Nixon.
Someone in Congress would need irrefutable evidence to bring down Obama, and even if they did, they would be subject to the utmost scrutiny. This person would be fighting against nearly all forms of media, and their reputations--if not impeccable--would be dragged through the mud while making their case.
Let's face it--what "evidence" would you show to indicate Obama wasn't a citizen, or was in college under a specific status? Any record, particularly an older, non-electronic one can be photoshopped or manipulated. "Witnesses" such as Obama's former roommate can be bullied, discredited, and otherwise tarnished to a point where they have little credibility.
I think that anyone in Congress would use proxies to come forth with any claims, and see if anything sticks. Unfortunately, the Obama boot-lickers will support him no matter what, offering any excuse in the world to not question their savior.