If I burn cards and I burn Flame Chompers, can I play it for through its effect?

No. Burning Flame Chompers does not trigger its ability — "discard" only means hand → trash (Rule 422.1), while Burn moves cards from the top of your Main Deck to trash (Rule 440.1). They're different game actions, so the "when you discard me" trigger never fires.

The card stays in your trash; you'd need a separate recursion effect to get it back. (Same reason it doesn't trigger when it's Killed.)