The House has jurisdiction over this question, not the courts. Given that they allowed the usurper to take power on January 20, 2009, I doubt that they would object to a patriotic American restoring the Office of the Presidency.
They might if that man threatened their gravy train in anyway.
From Wiki
In the legal system of the United States, the Supreme Court is the final interpreter of federal constitutional law, although it may only act within the context of a case in which it has jurisdiction.
NBC clause in our Constitution.
No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States.
The House is not allowed to reinterpret the meaning of the constitution. That is the pejorative of the Supreme Court
Again from Wiki
Under Marshall, the Court established the power of judicial review over acts of Congress,[10] including specifying itself as the supreme expositor of the Constitution (Marbury v. Madison)
The House can do whatever it wants but if the Supreme Court rules at candidate for POTUSA is not a NBC then that candidate is not eligible for the office and will not be placed on the ballot.
See same sex marriage for a recent example of the Supreme Court forcing it views on an unwilling Congress, an unwilling several states and an unwilling nation.