It honestly seems like there's only two ways to do it - overdose them (making them detonate spectacularly) or cause massive trauma to their heart. This would seem to make 'sense' (for a comic-based movie) - repairing the trauma and/or rebuilding a limb takes resources. If your heart isn't beating (like because it's GONE) those resources can't be delivered to the site of the trauma. Most likely, the heart (if the damage isn't too great) can be repaired, allowing for the restoration of the rest of the body.
When Iron Man blows a hole clean through one, he stays down - that hole went straight through where his heart would be.
When the woman in Tennessee is killed, it's by a massive explosion which almost certainly had enough concussive force to pulp her internal organs.
Later, both Pepper and Killian survive explosions. Pepper falls into a large explosion, and Killian is at the center of a suit detonation.
What Pepper fell into, though, was more firestorm than explosion by the time she reached it - she would have passed quickly through the worst of the heat, and into the area below it, where the heat would be less intense (though still fatal under normal circumstances). There was no massive concussive force of the explosion, only from the fall. Normal humans, depending on how they land, can survive a 200-foot fall (though it's rare). Given that Extremis could let someone walk essentially unscathed through fire, it's easy to see how Pepper could survive this.
Killian is more troublesome - he's at the center of the detonation of the Mark 42. The explosion probably should have killed him (I suspect the true reason he survived was plot armor). In-Universe, the explosion was likely intended to be a self-destruct, not an execution tool. It would have been focused on destroying the suit, though easily strong enough to kill any normal person inside. Thus, it's likely that the self-destruct used the dispersed power plants in the armor (which let pieces fly independently) to create explosions designed to destroy the tech. It would make sense for the focal point of these to be internal, focused outward. This is the only thing I could see that would have given Killian a chance to survive - most of the concussive force was directly outwards, not inwards. It still hurt like a son of a bitch, I bet.