A few of the more competitive residency programs do ask for your undergrad transcripts as part of your application. I haven't seen any residencies that ask specifically for MCAT scores, but since some undergrads do put your admission test scores (ie MCAT if you were pre-med) on your transcript, I guess they would might see it on there.
As was said, most residencies just ask for board pass letter (along with your pod school transcript and CV... quite a few also ask for LORs also).