George Romney was born in Mexico and was allowed to run for Pres in ‘68, so precsedent already set way before Obama
First of all, George Romney lost the nomination, but secondly, and more important, there is nobody who can "allow" or "not allow" someone to "run for President".
In fact, in the Constitution, there's no such thing as "running for President" (except in the incredibly poorly drafted XXII Amendment).
ANYBODY can "run for President". In fact, I believe that an Elector for President or Vice President can VOTE for anyone he likes, including a non-NBC, a ten year old, a resident of Indonesia, etc.
Someone who has RUN AND BEEN ELECTED but who is not QUALIFIED cannot be inaugurated and cannot exercise the powers of the office, but the sole judge and guardian of THAT is Congress, and, as with so many other of their functions, they have abdicated.