Old English dumb, of persons, "mute, silent, refraining from speaking or unable to speak," from Proto-Germanic *dumbaz "dumb, dull," which is perhaps from PIE *dheubh- "confusion, stupefaction, dizziness," from root *dheu- (1) "dust, mist, vapor, smoke," also expressing related notions of "defective perception or wits." Related: Dumbly; dumber; dumbest. Of animals, "lacking in speech," hence "without intellect" (c. 1200).
The fork in meaning probably comes via the notion of "not responding through ignorance or incomprehension." The Old English, Old Saxon (dumb), Gothic (dumbs), and Old Norse (dumbr) forms of the word meant only "mute, speechless;" in Old High German (thumb) it meant both this and "stupid," and in Modern German this latter became the only sense (the sense of "mute, speechless" being expressed by stumm). Meaning "foolish, ignorant" was occasional in Middle English, but the modern use in this sense (since 1823) seems to be from influence of German dumm, especially in Pennsylvania German.
dumb-cake ..., n. A cake made in silence on St Mark's Eve, with numerous ceremonies, by maids, to discover their future husbands. [Century Dictionary]
Applied to silent contrivances, hence dumb-waiter. Dumb ox "stupid man" is by 1756; dumb-bunny "stupid person" is college slang from 1922; dumb blonde "woman seen as incapable of comprehending anything complicated" is by 1936.