"to urinate, discharge the fluid secreted by the kidneys and stored in the urinary bladder," c. 1300, pissen, from Old French pissier "urinate" (12c.), from Vulgar Latin *pissiare, of imitative origin. To piss away (money, etc.) is from 1948. Related: Pissed; pissing. Pissing while (1550s) once meant "a short time."

He shall not piss my money against the wall; he shall not have my money to spend in liquor. [Grose, "Classical Dictionary of the Vulgar Tongue," 3rd edition, 1796]

To piss money on the walls "throw money around recklessly" is attested from 1540s.

He who once a good name gets,

May piss a bed, and say he sweats.

["Dictionary of Buckish Slang, University Wit and Pickpocket Eloquence," London, 1811] 

"vexed, irritated," c. 1400, figurative adjectival use of past participle of nettle (v.).

Nettled. Teized, provoked, out of temper. He or she has pissed on a nettle; said of one who is pevish or out of temper. [Grose, "Dictionary of the Vulgar Tongue," 1785] 

1958, intransitive, "go away," chiefly British; the transitive meaning "annoy (someone)" is by 1968, chiefly U.S.; from piss (v.) + off (adv.). Pissed off "angry, fed up" is attested by 1946 (Partridge says 1937); said to have been used in the military in World War II; in common use from 1970s.