No. If Obama is found to be ineligible, the de facto officer doctrine will apply. All appointments, executive orders, legislation signed by Obama, etc. will stand (remain in place).
The de facto officer doctrine confers validity upon acts performed by a person acting under the color of official title even though it is later discovered that the legality of that person’s appointment or election to office is deficient. Norton v. Shelby County, 118 U.S. 425, 440 (1886).
“The de facto doctrine springs from the fear of the chaos that would result from multiple and repetitious suits challenging every action taken by every official whose claim to office could be open to question, and seeks to protect the public by insuring the orderly functioning of the government despite technical defects in title to office.” 63A Am. Jur. 2d, Public Officers and Employees § 578, pp. 1080-1081 (1984) (footnote omitted).
I don't think the de facto doctrine applies to Barry:
"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
Maybe, maybe not. I’ve read otherwise, since the de facto officer doctrine may or does not apply to fraud, especially fraud that many people knew about at the time.
So it’s not really so much that he will be “found” to be ineligible, as much as he will be “revealed” to be ineligible, and that many, many people knew about it all along and indeed, committed crimes to get him where he is.
That is different than a regular old de facto officer doctrine.