Which is why the 20th Amendment says that a President elect - somebody who has received the electoral votes necessary to become President - cannot ACT as President if they have failed to qualify. The Constitution itself has the power to keep an unconstitutional President from taking any Presidential action, even if the people have elected that unconstitutional person. Same thing with a Vice-President elect.
And I would add this, also: Congress alone can remove a sitting President, but one of the reasons to impeach and remove someone is Constitutional disability - their inability to do the duties of the Presidency. If a court declared Obama unable to “act as President” because he failed to qualify according to the 20th Amendment’s requirements, Obama would not be able to do any Presidential actions, somebody else would have to act as President (and that would have to be determined by the court, to see if Obama ever WAS President since he was never eligible to be President), and Congress would presumably remove Obama because he is disabled from doing anything Presidential.
Congress would be the ones removing Obama, but they would do it after/because the courts had ruled him Constitutionally disabled from carrying out his Presidential duties.
The Obama thugsters have relied, all along, on the argument that it is nobody’s dang business if we have an unconstitutional Commander in Chief/President. And that is the insult of insults. As long as they screw us all equally it’s nobody’s business. Rape everybody, and then nobody can charge you with rape... And soldiers don’t suffer any particular damage from having a foreign enemy combatant as their CINC. By the time they’ve suffered “particularized harm” they are already dead and Obama is dancing over their graves.... According to the courts of the United States of America. (spit)