Normally med students (at least allopathic ones, I don't know about DO students) don't necessarily have to do away rotations to get into IM programs. However, in unusual situations (i.e. student from a med school with which a particular program like Mayo may not be familiar with...i.e. some DO schools or other med schools from outside the Midwest region, etc.) I think that if someone did an away there and did well, it might help with getting in. Of course, the opposite is also true (if you don't come across well, could likely kiss your chances there goodbye).
GI is a competitive fellowship to get into, and everyone I have known of who has gotten into Mayo or other similarly competitive academic GI programs has done research year(s), such as a postresidency clinical research fellowship, an MD/PhD degree, etc. There are always exceptions to this, but I'll bet a lot of them are residents from residencies like Harvard, UC San Francisco, etc.
However, whether or not you ever got into Mayo's GI fellowship, going there for residency would set you up well to get a fellowship somewhere.