"to dismiss lightly and contemptuously," literally "to turn aside with an exclamation of 'pooh,'" 1827, a slang reduplication of dismissive expression pooh. Among the many 19th century theories of the origin of language was the Pooh-pooh theory (1860), which held that language grew from natural expressions of surprise, joy, pain, or grief.
1590s, "a 'vocal gesture' expressing the action of puffing anything away" [OED], used as an exclamation of dislike, scorn, or contempt, first attested in Hamlet Act I, Scene III, where Polonius addresses Ophelia with, "Affection! pooh! you speak like a green girl, / Unsifted in such perilous circumstance. / Do you believe his tenders, as you call them?" But the "vocal gesture" is perhaps ancient.
"leader who maintains excessive bureaucratic control," 1888, from Pooh Bah, the name of the "Lord High Everything Else" character in Gilbert and Sullivan's "The Mikado" (1885).