Some of you guys. Honestly.
As an Agnostic, I could say that all religions are full of "dumb rules," but that doesn't make them any less important to the people who hold the beliefs. And it doesn't make those beliefs - or those people - any less worthy of respect. Medicine is not all about what's comfortable or logical to the people who practice it.
I've been asked to step out and find a female tech to do a 12-lead EKG on a female Muslim patient. Intellectually, yeah, I think it's dumb to consider all physical contact to necessarily have a sexual component. But I don't take it personally, and I don't imagine that my opinion matters more than the patient's just because I work there.
Same goes for a staff or faculty member, or even a student rotating through the department. TruckGirl, to answer your question, I'd think this student was observant of the customs of their religion, and that's all I'd think. If it had a negative effect on their grading, they might very well have a good case for lodging a complaint against the school. It's against the law to treat a student differently because of their religion, and whether a student shakes hands with staff has diddly to do with their ability to do the doctoring thing. So I'm with StinkyCheese and Seashelley on this.