He died from the wounds that he got from fighting Tigerstar after Tigerstar died.
He was actually hit by a burning tree that fell after he beat Tigerstar.
No, he died from his wounds. one, the tree was the Warriors' den, I'm pretty sure. And two, the fire didn't touch him, and three, it even says he died of his wounds on Warriors Wiki, and the Erins' confirmed it.
The exact quote is "she looked up as a bolt of lightning struck the fallen beech beside Firestar. The tree exploded into flame. Smoke rolled over Firestar."
The fallen beech is the Warriors' den. That's why it says 'fallen', if it fell on him, it would have mentioned it 'falling'.
Ashtorm's right. He died from wounds and lost his life before that the same way. -Fiercestar