I don’t think any precedent remotely exists from this.
On the one hand my feeling has always been that since Obama was part of fifty state elections and on fifty state ballots, the time and place to verify his eligibility to hold office was back then, when he was put on those ballots.
Due to Federal supremacy, the States have to follow the Constitution when putting individuals on the ballot. So the individual States can’t have 21 year olds or non-natural born citizens on the ballot. However, if they went ahead and put one on the ballot anyway, I would think a citizen of that State would have to sue in Federal court to block the action (at which point the SCOTUS/lower level Federal courts would note the obvious ineligibility of the candidate and order he be removed from the ballot.)
That didn’t happen, or rather if anyone did actually sue to prevent Obama from staying on the ballot, those cases did not go anywhere.
So once election day was past, all the State electoral votes were sent to the “Seat of Government” and certified by the President of the Senate. The individual who had a majority of EVs then became President, that is exactly what happened.
I would imagine that if a concerned citizen felt that Obama was not a natural-born citizen, he could have filed suit against the President of the Senate to stop their certifying the results until the natural-born citizen issue was resolved. That didn’t happen, or if it did the suit didn’t go anywhere.
Once those results are certified though, and especially once Obama was sworn in, I do not know that there is any mechanism aside from impeachment that could remove Obama from power.
The Constitution sets rules as to who can be President, and it clearly says anyone who is not a natural born citizen is ineligible to hold the office. At the same time, there is absolutely no precedent for a President just being declared ineligible and removed, for example let’s say we had 100% definitive proof Obama was born in Kenya. Someone proved this in court and it made it up to the SCOTUS level, the SCOTUS ruled Obama was “no longer President.” Well, I don’t know that the SCOTUS can do that, because even if it was 100% certain Obama was ineligible to be elected, once sworn in there is absolutely no constitutional support for the concept of the SCOTUS being able to unilaterally remove a sitting President, for any reason.
Congress could, as always, impeach him and remove him from office. The impeachment power is entirely discretionary, though, and Congress would be under no obligation to impeach him.
As far as the validity of the laws Obama signed, as the properly sworn in President, they would all still have full force of law and persons who ignored their validity would almost certainly find themselves in serious legal trouble.