You attack with a flaming long sword non-lethally taking the -4 penalty to hit, and succeed in damaging the target.
The target has no damage reduction, so you definitely (for the sake of argument) deal 1d8 non-lethal damage. And.... any more? Is the 2d6 fire ignored, dealt as non-lethal (pretty ridiculous) or dealt lethally?
The only mention I can see about this kind of interaction is under descriptions for sneak attack which says (text from the rogue class shown here):
...She cannot use a weapon that deals lethal damage to deal nonlethal damage in a sneak attack, not even with the usual –4 penalty.
Which in itself I find kind of unclear. Is attacking non-lethally against a flatfooted opponent impossible for a rogue? Or does it just mean that if you want to attack non-lethally you cannot add sneak attack dice? (Probably the latter)
To clarify: The core question here is whether magic weapons that add damage also have their extra damage converted to non-lethal damage when used for a non-lethal attack. The section on rogue sneak attacks is merely an example of the apparently minimal rules directly addressing this topic.