The one that says “Burn the Heart of a Destroyer” wants you to burn a Destroyer Lodestone.
However, I’m stuck on the one that says “Burn the Heart of the Megadestroyer”. I got the one that’s used in making Mawdrey, but I couldn’t burn it, and his World Boss chest didn’t drop one that I could burn.
I’m going to try removing the Mawdrey one from my inventory next time I fight him to see if it was just a glitch with dropping.