He is not eligible to assume the presidency under Art II Section 1. That is the basis for any action. Especially since everyone else is cowards.
I took an oath to protect and defend the constitution against all enemies foreign and domestic. That does not make this country a third world one. Our oath is unique in the military structure of the world and precludes ever becoming a third world nation.
That would be the basis for taking action under Article I, Sections 2 and 3.
Say it were shown beyond a reasonable doubt that Zero was actually born in Kenya to BHO, Sr., and a trollop from Liverpool. That would make him ineligible for sure. Where would the case end up? Not the Supreme Court. Under Article I, they don't have the authority to remove him. It would be up to the House and Senate, solely.
Of course, if the Congress didn't up and oust him pronto, maybe a general contemplating his oath might feel compelled to act.
But the above is strictly hypothetical, since, since despite rumors to the contrary, Zero was actually born in Honolulu.