Old English bitan "to pierce or cut with the teeth" (class I strong verb; past tense bat, past participle biten), from Proto-Germanic *beitanan (source also of Old Saxon bitan, Old Norse and Old Frisian bita "cut, pierce, penetrate," Middle Dutch biten, Dutch bijten, German beissen, Gothic beitan "to bite"), from PIE root *bheid- "to split," with derivatives in Germanic referring to biting.
To bite the bullet is said to be 1700s military slang, from old medical custom of having the patient bite a lead bullet during an operation to divert attention from pain and reduce screaming. Figurative use of this is from 1891; the custom itself attested from 1840s.
Figurative bite (one's) tongue "refrain from speaking" is by 1590s; to bite (one's) lip to repress signs of some emotion or reaction is from early 14c. To bite off more than one can chew (c. 1880) is U.S. slang, from plug tobacco.
To bite the dust "be thrown or struck down," hence "be vanquished, die, be slain, perish in battle" is from 1750, earlier bite the ground (1670s), lick the dust (late 14c.), which OED identifies as "a Hebraism," but Latin had the same image; compare Virgil's procubuit moriens et humum semel ore momordit.