Pictograph of a person with an open mouth (欠) drinking from a wine vessel (酉). 欠 also represents the sound. The 酉 (wine) component and the open mouth was later reanalyzed as 飠 (eat).
Depicts a person bending over with their mouth open.
飲 and 欠 don't sound similar in modern Mandarin due to historical phonetic changes. They were more similar in older Chinese.飲 /*q(r)[u]mʔ/, /*q(r)[ə]mʔ-s/ 欠 /*[k]ʰ(r)om-s/