The Court would not be ordering the removal of a President, but rather a usurper. They would first be saying that he was never elibigle, and then if necessary and probably in a latter case, order his removal. Not that it's likely to be necessary, if the Supreme Court should ever declare him to be ineligible. Someone would either persuade him to step down, or it would be torches and pitchforks time. (although I'd pretty a nice rifle myself). If he's ineligible, who is going to defend his sorry ash, the military? Nope their oathers are to the President, not some resident usurper. The Secret Service? Same for them, they defend the President.
The Constitution provides that the President may be removed by impeachment and conviction in the Senate, it says nothing about removing a usurper. I think the founders figured the Second Amendment covered that potentiality.
Well, that's the key, isn't it?
Absent torches and pitchforks, the Court does nothing or at most opines and defers to an impeachment process that doesn't happen. But should torches and pitchforks materialize for real, the Congress impeaches and removes, so the Court needn't do anything. Either way, the Court is at most an adviser, not a remover.
The key is to get up the torches and pitchforks. Counter-factual (not born in Hawaii) or esoteric (all the Vattel stuff) arguments about eligibility won't do the job. What Obama is actually doing in office just might.