"Saying Goodbye" from The Muppets Take Manhattan is a real tear-jerker, but on TMS, I'd say Gonzo's version of "My Way" (in the Lola Falana episode when he leaves the show for an acting career in India) and "When I Grow Too Old to Dream" (from the Linda Ronstadt episode) are right up there, too.
Honorable mention: the version of "When the River Meets the Sea" that Jerry Nelson and Louise Gold performed at Jim's memorial (if memory serves correctly).
YAAAYY!!! 1000th post on this thread.