When Flame Chompers die, can I pay a Fury Rune to play it?

No. Flame Chompers' ability triggers only on discard (hand → trash), not on being Killed (board → trash). Those are different game actions with different trigger conditions.

The card reads: "When you discard me, you may pay fury rune (power) to play me." Being destroyed in combat, removed by a Kill effect, or any other form of death from the board does not satisfy "discard" — Discard specifically means moving a card from hand to trash. You would need a separate recursion effect (like Unlicensed Armory or similar) to bring Flame Chompers back from the trash after it's Killed.