Well, not exactly. First, if he's not eligible, he's not a sitting President, he's an illegal usurper. Hard to say who would have the power to remove him, should he not go peaceably. But Congress has no explicitly delegated power to remove a usurper, at least I can find no such power in Art 1 section 8, or anywhere else in the Constitution, so they could not delegate such a power to the DC court, or anyone else. It's possible that such a power is "hidden" in the last clause of section 8.
To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers, and all other Powers vested by this Constitution in the Government of the United States, or in any Department or Officer thereof.
The Courts probably have the power under Article IV's second clause:
This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding.
Along with article III section 2
The judicial Power shall extend to all Cases, in Law and Equity, arising under this Constitution,
“In the United States the right to a public office is tried by quo warranto or similar procedure, regulated by the state laws. Proceedings by quo warranto lie in a United States court for the removal of persons holding office contrary to art. xiv. s. 3 of the Amendments to the Constitution (act of the 31st of May 1870, c. 14).”
From the following site:
http://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Quo_Warranto