Look at 8.b-c in
Advisory Circular 90-108 (bottom of page 5 to top of page 6). Generally, you need the equipment named in the title and other notes of an approach. So, for a VOR/DME 15 approach, your plane needs both a VOR and a DME. Advisory Circular 90-108 is what allows a "suitable RNAV system" (GPS) to substitute for other navigation equipment.
However, it specifically disallows substituting for
lateral guidance in the
final approach segment. Normally this is not a problem, as the most common substitution is GPS for DME, and in almost all approaches, DME is there for step-down fixes or other progression-type (not lateral) guidance. Similarly, most VOR/DME approaches that do include arcs (where the DME provides lateral guidance), those arcs are not part of the final approach segment (for example,
VOR/DME 22 at HUT).
The odd thing about VOR/DME 15 at MTN is that the DME distance is providing your lateral (turn right / turn left) guidance through the final approach segment. For that reason, it can't be substituted by GPS.
The same is true for standard VOR approaches: you must monitor a real-live radio VOR during the final approach segment (assuming the VOR radial serves as lateral guidance for the approach, as is typical). Also, you can never substitute for a localizer.
Oddly enough, with a certified GPS, working DME, but missing or broken VOR radio, you may fly the MTN VOR/DME 15, substituting GPS for the VOR step-down fixes (as they aren't lateral) and using the real-life DME for lateral guidance. Good luck finding a suitable alternate without a VOR though...