We have a good idea what the left would do if it was undeniable he was foreign born.
First, politicians would enact legislation to allow foreign born citizens to be president, grandfathering in Obama if needed. We know this from attempts of numerous Democrats having put forth legislation to this effect prior to and after the One was chosen.
The Courts would rule the people voted him in and would be hesitant to remove him from office because of this - that the people were lied to would not matter since politicians lie on a regular basis and that does not change the outcome of an election.
The press would put forth editorials on how the US is missing out by not allowing the greatest minds or most qualified people into the White House because they were born on foreign soil. They would defend Obama throughout if they even lowered themselves to report the story. Birthers would be blamed for harming Michelle and the Obama’s wonderful children.
The left would split with 1/2 denying he was foreign born even if he admitted to it, the other 1/2 would admit he was but not care. We’re all children of the world donchaknow...
Some law professors have already written papers arguing this very thing. They seem, however, to see access to the Presidency as a civil rights issue, when in fact it is a matter of national security that the chief executive be a natural born citizen with no conflicting loyalties.