Chewed bones are obtained exclusively as a drop from mithril dragons in the Ancient Cavern. No other monsters in the Ancient Cavern will drop chewed bones, and looting the skeletons won’t give chewed bones either (only mangled bones).
When these bones are put on a pyre ship and burnt (requires a tinderbox, axe and any type of non-pyre log), they will summon a Barbarian spirit, who may thank you for putting his soul to rest with an item. The reward may be a dragon full helmet, various potions, runes or projectile ammunition. Many players will attempt killing mithril dragons in the hopes of obtaining a dragon full helmet drop; if unsuccessful, they can take any gained chewed bones to be burned in a pyre ship for another chance at gaining a dragon full helmet, however the chances of that are fairly rare. Note that using higher level logs does NOT increase chances of receiving a dragon full helm.
Burning chewed bones via a pyre ship provides an experience bonus (300%) to any bones buried for the next 5 bones. You will receive a message saying, “You gain extra prayer experience as the ancestral barbarian spirits watch over you.” This bonus accumulates. Burning 2 chewed bones consecutively provides a bonus for burying twice as many bones as burning one chewed bone. Dying does not remove the accumulation of bonuses.