He wouldnt have to be impeached. He’d be forced to resign and since he was never a legal president, anything he or his administration did would be void. Top it off, Biden would not be able to be sworn in.
I think the most likely scenario is that both political parties would pretend Obama's resignation had nothing whatsoever to do with eligibility fraud. Biden gets in and he, Boehner, and Reid link arms in a rose garden press conference, promising a national unity government to get the country through this time of national trauma.
There is no way to resign from an position you do not legally occupy.