in golf, 1721, back-formation from teaz (1673), taken as a plural; a Scottish word of uncertain origin. The original form was a little heap of sand. The verb meaning "place a ball on a golf tee" is recorded from 1673; fig. sense of "to make ready" (usually with up) is recorded from 1938. Teed off in the fig. sense of "angry, annoyed" is first recorded 1953, probably as a euphemism for p(iss)ed off.
Start or begin, as in We teed off the fundraising drive with a banquet. This usage is a metaphor taken from golf, where
tee off means “start play by driving a golf ball from the tee.”
[ Second half of 1900s
Make angry or irritated, as in That rude comment teed him off, or I was teed off because it rained all weekend.
[ ; mid-1900s