In order for a Bill to become law, it must be signed by the President. Ditto an Executive Order. If the maid signed it, it would not be law. If he is proven ineligible, he cannot be POTUS, he could not have been POTUS, and nothing he signed could have the weight of law any more than if the maid had signed it.
Ten days after presentation of the bill to the President, if the president does not veto the bill, the bill becomes law.
So, no, the President doesn't have to sign it.
How does the court deal with a fraudulent President?
It may be that, say, the Health Care bill is deemed to have never been signed by a legitimate president and so became law after ten days.