In a game relying less on monsters and treasure, you may need alternate methods of advancement. A simple solution is to just use the existing system in any event.
Treat each puzzle, trap or challenge as a monster for experience purposes. A simple puzzle may be a level 2 threat, while a devious one may be level 10.
A trap that can slow down the characters might be level 3, but additional defenses or tricks surrounding it (such as being hard to disarm) can increase the amount.
The same system can be used for accomplishing various quests and tasks, though here, a fixed XP award (equivalent to treasure) may be more appropriate, as in a non-combat oriented game, this will be the main source of experience.