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1933, from phrase to speak with one's tongue in one's cheek "to speak insincerely" (1748), which somehow must have been suggestive of sly irony or humorous insincerity, but the exact notion is obscure.
Old English tunge "organ of speech, speech, language," from Proto-Germanic *tungon (cf. Old Saxon and Old Norse tunga, Old Frisian tunge, Middle Dutch tonghe, Dutch tong, Old High German zunga, German Zunge, Gothic tuggo), from PIE *dnghwa- (cf. Latin lingua "tongue, speech, language," from Old Latin dingua; Old Irish tenge, Welsh tafod, Lithuanian liezuvis, Old Church Slavonic jezyku).
For substitution of -o- for -u-, see come. The spelling of the ending of the word apparently is a 14c. attempt to indicate proper pronunciation, but the result is "neither etymological nor phonetic, and is only in a very small degree historical" [OED]. Meaning "foreign language" is from 1530s. Tongue-tied is first recorded 1520s.
"to touch with the tongue, lick," 1680s, from tongue (n.). Earlier as a verb it meant "drive out by order or reproach" (late 14c.). Related: Tongued; tonguing.
A mobile mass of muscular tissue that is covered with mucous membrane, occupies much of the cavity of the mouth, forms part of its floor, bears the organ of taste, and assists in chewing, swallowing, and speech.
Ironically: “The critic's remarks of praise were uttered strictly tongue-in-cheek.”