c.1200, furen, figurative, "arouse, excite;" literal sense of "set fire to" is from late 14c., from fire (n.). The Old English verb fyrian "to supply with fire" apparently did not survive into Middle English.
The sense of "sack, dismiss" is first recorded 1885 in American English (earlier "throw (someone) out" of some place, 1871), probably from a play on the two meanings of discharge: "to dismiss from a position," and "to fire a gun," fire in the second sense being from "set fire to gunpowder," attested from 1520s. Of bricks, pottery, etc., from 1660s. Related: Fired; firing. Fired up "angry" is from 1824. Firing squad is attested from 1904.