Some of the non-combat RPG things I've enjoyed quite a bit:
Trying to negotiate a treaty in a tense diplomatic situation.
SE learning that his character was actually the lost child of a God we'd been dealing with for a while.
Mistress of Darkness falling in love with every male NPC in the game.
Non-combat scenes from books and movies that I find fascinating:
The dinner scene early in Dune has more intrigue and strategy than any battle scene I've ever read. Just get a bunch of NPCs who all want to kill each other and the PCs, but have to do it covertly, and you've got a lot of potential. You could swap out the dinner for a cocktail party, a royal ball, or even a masquerade of some kind--anything where the character have to remain civil and use social skills instead of combat skills.
The Poseidon Adventure put the characters in a normal situation that was completely backward: a cruise ship turned upside down, and slowly sinking. They had to navigate through a wrecked boat, avoiding fires and the ever-rising water level, but they had to do it in a situation where none of the stairs, ladders, or rooms worked the way they were supposed to (ie, the door out of the ballroom was on the ceiling instead of the floor).
In the movie The Fugitive, Dr. Kimball has to do a lot of unconventional stuff to track down the killer, because he doesn't have access to any conventional forms of research: he had to talk to old friends, sneak in and out of hospitals, and come up with weird cover stories when he called people to keep them from becoming suspicious. Typical RPG research is really easy: you look up a bad guy in a database and get his picture, his important info, and so on; Kimball can't do any of that, so he had to actually go into the prison and request an audience with the guy, made all the more difficult because he was a wanted man. Denying the characters their typical resources can make something as simple as recon into a very interesting situation.
I know there's more, but that's all I have time for right now. Anybody else have a favorite scene that could provide cool non-combat RPG fodder?