The only thing even marginally resembling a bone that might get thrown the Birthers' way would be some sort of legislation requiring that the President-elect and Vice President-elect submit documentation to Congress that they meet the qualifications for the office. Congress' authority to verify that the President-elect and Vice President-elect qualify for office is certainly implied in the original Constitution, and spelled out in the 20th Amendment. Since Obama would have no problem meeting such a requirement, it would be a meaningless bone.
Certainly no hearings into whether Obama really is qualified to be President.
even that would cause a controversy over separation of powers for no reason.