tip (v.1)
c.1300, "to knock down, overturn, topple, knock askew" (transitive), of uncertain origin, possibly from Scandinavian (compare Swedish tippa "to tip, dump"), or from a special use of tip (n.). Intransitive sense of "to fall over, be overturned" is from mid-15c. Related: Tipped; tipping. To tip the scales at "weigh (so much" is from 1879. Tipping point attested by 1972. To tip (one's) hand "reveal one's intentions" is from 1907, an image from poker-playing.
tip (n.1)
c.1400, "extreme end of something, top of something round or pointed, metal attachment to the end of something," from Middle Low German or Middle Dutch tip "utmost point, extremity, tip" (compare German zipfel, a diminutive formation); or from a Scandinavian source (compare Old Norse typpi).
tip (v.2)
"give a small present of money to," c.1600, originally "to give, hand, pass," thieves' cant, perhaps from tip (v.3) "to tap." The meaning "give a gratuity to" is first attested 1706. The noun in this sense is from 1755; the noun meaning "piece of confidential information" is from 1845; and the verb in the sense "give private information to" is from 1883.

The word's supposed origin as an acronym is highly unlikely and the story seems to be no older than an editorial in "Life" magazine from July 15, 1946, claiming the restaurant server's word tip "probably comes from a London coffeehouse custom of two centuries ago when the words 'To Insure Promptness' were written on notes to the waiter, with coins attached. Later just the initials T.I.P. were used." There is no historical evidence for this. Also see here.
tip (v.3)
c.1200, "to strike, occur suddenly," of uncertain origin, possibly from Low German tippen "to poke, touch lightly," related to Middle Low German tip "end, point," and thus connected to tip (n.); or else related to tap (v.1). Meaning "strike sharply but lightly" is from 1560s. Sports sense is from 1816, originally in cricket. Related: Tipped; tipping.
tip (n.2)
"a light, sharp blow or tap," mid-15c., from tip (v.3).
tip (v.4)
"put a tip on, adorn with a tip," late 14c., from tip (n.) or Old Norse typpa. Related: Tipped; tipping.