Chewed bones are obtained exclusively as a drop from mithril dragons in the Ancient Cavern at a rate of 3/128. 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, blisterwood logs do not work), they will summon a Barbarian spirit, who may thank the player for putting his soul to rest with an item. The reward may be a dragon full helmet, various potions, runes or projectile ammunition. Players may kill 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. 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 bones, the amount varying by logs burnt. The player will receive a message saying, "You gain extra prayer experience as the ancestral barbarian spirits watch over you." This bonus accumulates and is not lost upon death. Burning 2 chewed bones consecutively provides a bonus for burying twice as many bones as burning one chewed bone. The Bonecrusher doesn't automatically crush chewed bone drops. Burning Chewed bones on a pyre ship constructed with magic logs is a requirement for the Kandarin Elite Diary.
An ancient barbarian spirit being laid to rest
The following drop rates are provided by Jagex, unless otherwise specified.
For an exhaustive list of all known sources for this item, see here