Legally, that's just not true. Obama would not be the first government official in US history who has been discovered to be not properly holding his office by reason of a lack of some sort of eligibility or qualification.
In fact, it has happened from time to time at all levels of government and the law has become well established over the centuries as to what happens under the "de facto officer" doctrine.
Under this doctrine, even though it is later discovered that a government official who holds color of official title to his office by virtue of a known election or appointment failed to meet a prerequisite of holding office, such as an eligibility requirement, nevertheless actions taken by him within the scope and by the apparent authority of that office will be considered valid and binding. This has consistently been the position taken by the courts.
The only difference in this case is that it is the president. And in order to remove him, under the Constitution the Congress would have the responsibility to determine in an impeachment trial whether he is wrongly holding his office, and if so, to remove him from office after which the courts would then have jurisdiction to try him for crimes.
I call BS on your de facto officer doctrine read the case law.....it does not apply to Obama...