c. 1200, furen, "arouse, inflame, excite" (a figurative use); literal sense of "set fire to" is attested from late 14c., from fire (n.). The Old English verb fyrian "to supply with fire" apparently did not survive into Middle English. Related: Fired; firing.
Meaning "expose to the effects of heat or fire" (of bricks, pottery, etc.) is from 1660s. Meaning "to discharge artillery or a firearm" (originally by application of fire) is from 1520s; extended sense of "to throw (as a missile)" is from 1580s. Fire away in the figurative sense of "go ahead" is from 1775.
The sense of "sack, dismiss from employment" is recorded by 1877 (with out; 1879 alone) in American English. This probably is a play on the two meanings of discharge (v.): "to dismiss from a position," and "to fire a gun," influenced by the earlier general sense "throw (someone) out" of some place (1871). To fire out "drive out by or as if by fire" (1520s) is in Shakespeare and Chapman. Fired up "angry" is from 1824 (to fire up "become angry" is from 1798).