Believe everything Obama put his signature to, while acting as President would be declared null/void/invalid. Think if he were removed; there would be a race war (which may be inevitable, irrespective of Obama's status and so a 'non-argument' really).
Think I would leave Obama in White House/with family; as a fully recognized 'poser' President; a 'figure head' and nothing more; while authentic Leadership straightened out the paper work until official Election .
In truth and more than likely; NOTHING would be done; until next election; where, then, what damage could be undone would be. Depending on 'who' is in White House, that is. . .
Certainly, this man as life-long 'poser'- is known to many in Washington and yet, all remain on the path, of least resistance. (Or so it appears; albeit we do have a few beginning to step - just barely - outside it.)
In order to mount a successful attack on his status, you need to get someone who has legal standing; and they need to be represented by competent counsel who is prepared to force the factual argument and appeal adverse trial court rulings.
That generally means to commence a legal proceeding against some action he has taken which requires a President to implement.
One good example is the Military Officer who can object to an order. So far, we haven't seen anyone with adequate resources and competent counsel.
We are setting up for another good combination of facts. Appointing two extreme Liberals to the Supreme Court leaves you with the ability to forecast their votes on many issues. So in most cases, a party will not risk his case by seeking to disqualify them from the panel hearing the case--he knows how they will vote.
A Supreme Court Justice must be appointed by a "President"--these two ladies were not because the person who appointed them is not eligible to hold the office under the Constitution.
What would be the consequences? Would he be removed and if so by whom?
No. It’s too late. About all it might do, is prevent him from another term