Not so. Because of the de facto officer doctrine, Obama's removal from office would not result in the removal of his appointees. In any event, the Supreme Court denied certiorari in another eligibility case last month or so, and Kagan and Sotomayor did not recuse.
--------------------------------------------------------
I'm not so sure your opinion there on the de facto officer doctrine is a 100% shoe in.
"To satisfy the doctrine, the officer must be in the unobstructed possession of the office and discharging its duties in full view of the public, in such manner and under such circumstances as not to present the appearance of being an intruder or usurper."
http://info.libraries.vermont.gov/SUPCT/160/op92-113.txt
Barry has all the "appearance" of being an "intruder" to the office.
Does mention of the possibility in a petition amount to a specific request? A specific request would have been better.