early 12c., brennen, "be on fire, be consumed by fire; be inflamed with passion or desire, be ardent; destroy (something) with fire, expose to the action of fire, roast, broil, toast; burn (something) in cooking," of objects, "to shine, glitter, sparkle, glow like fire;" chiefly from Old Norse brenna "to burn, light," and also from two originally distinct Old English verbs: bærnan "to kindle" (transitive) and beornan "be on fire" (intransitive).
All these are from Proto-Germanic *brennanan (causative *brannjanan),source also of Middle Dutch bernen, Dutch branden, Old High German brinnan, German brennen, Gothic -brannjan "to set on fire;" but the ultimate etymology is uncertain. Related: Burned/burnt (see -ed); burning.
Figurative use (of passions, battle, etc.) was in Old English. Meaning "be hot, radiate heat" is from late 13c. Meaning "produce a burning sensation, sting" is from late 14c. Meaning "cheat, swindle, victimize" is first attested 1650s. In late 18c, slang, burned meant "infected with venereal disease."
To burn one's bridges (behind one) "behave so as to destroy any chance of returning to a status quo" (attested by 1892 in Mark Twain), perhaps ultimately is from reckless cavalry raids in the American Civil War. Of money, to burn a hole in (one's) pocket "affect a person with a desire to spend" from 1850.
Slavic languages have historically used different and unrelated words for the transitive and intransitive senses of "set fire to"/"be on fire:" for example Polish palić/gorzeć, Russian žeč'/gorel.