Technically it would not be a case of removing the President. For no person not eligible by the criteria of the Constitution can actually *be* President. The Court would merely declare him not to be eligible, and thus not the President.
An honest man would then step down, but then an honest man would not run or take office knowing he was not eligible. Some have suggested that the provisions of the XXVth amendment concerning a President not able to exercise the duties of the office might be applied, but then again if he wasn't the President, they might not be applicable.
The problem is of course that the Constitution doesn't contemplate an ineligible person becoming de facto President. But that does not, and cannot, mean that he would get to continue exercising the powers of the office. That would make a mockery of the Constitution.
It's possible that once the Supreme Court declared him to be ineligible, the Congress would step up to the plate, impeach and remove him from office. In fear of their own re-election if for no other reason. Of course impeachment is for Presidents not usurpers, so that might be problematic as well. They might instead have to pass a special law or resolution.
You've left out Justice Roberts in this whole debacle. Does he have to recuse himself because he actually swore the Barry into office? It's tough to listen to a case that says the President isn't really the President, when you're one of the key figures involved in his installment in the office.
"The problem is of course that the Constitution doesn't contemplate an ineligible person becoming de facto President."
This above all others, is the primary problem, at least legally. Wherever it is we are today, no one - included the Framers - ever anticipated that we would be here.
the Congress would step up to the plate, impeach and remove him from office.
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How can the Congress impeach someone who is not president. I would think he would simply be arrested for the crime of massive fraud and frog marched out of the White House.
I can envision the scenario where the already dishonest DNC hold a mock hearing in the Senate Judiciary and proclaim Obama ineligible on a technicality, then move to swear in Joe Biden as President immediately. Covering for their criminality will be the primary motivator for PelicaNazai and RancidReid.