[centurion316]

The courts do have the power to declare him ineligible. If he's ineligible, no one need obey his orders. No military, no civilian. He might himself put out on the street by the Secret Service. If he wasn't President, he couldn't even qualify for a clearance as White House gardener

Where did you get this idea? You can remove a President in one of four ways: death (Lincoln, Kennedy, etc),resignation (Nixon), impeachment (None - Johnson and Clinton tried but found not guilty), and defeat in an election (G.H.W. Bush, Carter, others). Absent removal by one of these legal/natural means, the President remains in Office and retains all the powers of that Office. What court do you think has the power that you suggest?

[El Gato]

Where did you get this idea?

Where does anyone get the idea that someone ineligible to the office of President can ever be President?

You can remove a President in one of four ways:

But if he's not eligible, he's not President, and never was. Elections, oath taking, etc aside. No one, No One, has certified his eligibility, only his election. The votes were properly cast, but properly cast electoral votes cannot make someone who is not eligible, suddenly eligible. Impeachment is for President, not ineligible usurpers.

The courts get their powers from Art. III of the Constitution, which says in part:

You can remove a President in one of four ways:
...
Section. 2. The judicial Power shall extend to all Cases, in Law and Equity, arising under this Constitution, the Laws of the United States, and Treaties made, or which shall be made, under their Authority;

This would be a case arising under Article II of the Constitution, and thus the federal courts have jurisdiction to decide it.