Does quantum mechanics play a role in protein folding?
There's several aspects to your question. 

Does quantum mechanics play a role in protein folding? Yes. The origin of the van der Waals interaction is ultimately a quantum mechanical one. (At least the induced dipole portions: how the electrons move with respect to each other and an external electrical field is driven by quantum mechanics.) Also, while part of hydrogen bonding can be described by classical dipole-dipole interactions, there's a non-negligible portion which is due to non-classical electron interaction. (How much this last bit contributes, if at all, is a bit controversial, though.)

But that's all electrons. The other part of your question is asking implicitly about the quantum behavior of nuclei.

In molecular modeling we often use a viewpoint termed "the Born-Oppenheimer approximation". That is, while we allow the electrons to delocalize and exchange and be their quantum mechanical self, we treat the nuclei pretty much like classical particles. Using this approximation, we can get pretty decent correspondence with experimental results.

So the best we can tell, the quantum nature of atoms as a whole (that is, nuclei as opposed to electrons) isn't needed for things like protein folding. The nuclei are probably not substantially delocalized, they're probably not interfering with one another, and they're probably not quantum tunneling.

Probably. There's some evidence that quantum tunneling of nuclei in the process of enzyme catalyzed reactions. This is mainly hydrogens, though. In most of these experiments the quantum tunneling drops off dramatically when you go to deuterium or tritium nuclei. I'm unaware of any system where carbon/oxygen/nitrogen tunneling is seriously proposed - they're just too heavy.

So there may be some contribution to the quantum nature of nuclei to protein folding, but all indications at this point indicate that it would be minor. The quantum properties of electrons, though, are definitely important.