It wouldn't be difficult to learn the fact, but producing some evidence would be the hurdle. Clerks and bureaucrats at all levels have access to records like birth certificates. They can easily view them, just not release them.
That said, I'm skeptical. He would have had to produce a birth certificate to obtain a passport which he would need to travel internationally. He has travelled as a Senator, hasn't he?
Couldn't he have obtained a passport even if he were naturalized?
Well, technically, a Consular Report of Birth Abroad is a legitimate substitute for a United States birth certificate for purposes of demonstrating citizenship, but I don't think anyone has ever suggested that Senator Obama has such a document since he was ostensibly born in the United States.