The act of using an ability that has a long 'recharge time'. When a Hearthstone has just been used, and is in cooldown, a player is said to have 'burned his Hearthstone', until the cooldown period completes.
Believed to have originated with the massively-multiplayer game 'Guild Wars' (but maybe not, since that game is relatively young in age).
An alternate definition came about in Everquest. Burning was to dps down a mob as fast as possible, usually using inefficent spells and abilities to save time.