Old English bitan (class I strong verb; past tense bat, past participle biten), from Proto-Germanic *bitan (cf. Old Saxon bitan, Old Norse and Old Frisian bita, Middle Dutch biten, Dutch bijten, German beissen, Gothic beitan "to bite"), from PIE root *bheid- "to split, crack" (see fissure).
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 from 1891; the custom itself attested from 1840s. To bite (one's) tongue "refrain from speaking" is 1590s. To bite the dust "die" is 1750 (Latin had the same image; cf. Virgil: procubuit moriens et humum semel ore momordit). To bite off more than one can chew (c.1880) is U.S. slang, from plug tobacco.