If it is related to a crime having occurred within the Law Enforcement Agencies jurisdiction, I believe the Hospital would have to answer to a judge if they refused, but I don't see any angle where they can be compelled in this case. Perhaps if enough information of a crime is uncovered, it might be possible to force their hand, but till then, I would be very surprised if they didn't ignore him.
I guess Maricopa county will issue a proclamation of sorts, detailing why Obama is not allowed on the ballot. Then the case can go to Arizona's supreme court, and force all of Arizona to remove him. Then on to the US supremos, and these people at the Hospital can explain themselves in front of that court. That could be embarrassing for them if they in fact know he was not born there and refused to let it be known .