Do flaming (and other elemental damage) weapons that have the trip special weapon feature deal their extra damage when they are used in a trip attack and hit successfully?
No, they do not.
The fire (or other elemental) damage dealt by such weapons is extra damage, by definition added to whatever damage the weapon deals under other circumstances. A trip maneuver does not deal damage, and thus does not deal the extra damage either.
This is made explicit under Actions in Combat:
options that increase damage don't cause attacks to deal damage if they wouldn't otherwise do so (such as Vital Strike and trip)