How about if he applied for financial aid in college as a "foreign" student? Would he be able to do that as a US citizen? I don't know that he did - no one knows, really. I mean, those records have not been made public for some reason.
IF that happened (and that rumor started from an April Fools joke with a fake student ID image), it would just be fraud, that wouldn't renounce his citizenship.
http://www.snopes.com/politics/obama/birthers/occidental.asp(If you don't like Snopes, many sites found the same thing and that article links to other sources of proof it was a hoax).
Occidential even made a statement they didn't give scholarships to foreign students, ditto to the Fulbright scholarship he received.
Again, all internet static. In this case, 'common knowledge' started out as a hoax making fun of birthers.