late Old English, adumbian, "to become mute, be silent, keep still," from dumb (adj.). From c. 1600 as "to make mute." Related: Dumbed; dumbing. To dumb (something) down "make less intellectually challenging, make simpler to understand" is from 1933.

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." The -b has probably been silent since 13c. 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.

also dumbass, "stupid, ineffectual person," by 1959, from dumb (adj.) "stupid" + ass (n.2).

also dumbbell, "one of a pair of weighted bars used for exercise," by 1785, earlier (from 1711), according to OED, an apparatus like that used to ring a church bell, but without the bell (hence dumb); used for physical exercise but sometimes also to practice ringing changes. See dumb (adj.) + bell (n.). If this is right, the word must have been transferred; earlier 18c. references make mention of "pulling" or "ringing" dumb-bells and note that it can be done only indoors. The following is a footnote to the 1903 reprint of Joseph Strutt's 1801 "The Sports and Pastimes of the People of England":

The origin of the term is somewhat curious. Dumb-bells take their name by analogy, as was pointed out in Notes and Queries in 1861, "from a machine used for exercise, consisting of a rough, heavy, wooden flywheel with a rope passing through and round a spindle ... and set in motion like a church bell." This statement, however, does not sufficiently explain the transference of such a name to the short bar and rounded lead or iron ends of a hand dumb-bell. This difficulty was explained by the late Chancellor Ferguson in a paper read before the Archaeological Institute in 1895, wherein a dumb-bell apparatus, now at Lord Sackville's seat at Knowle, was described and illustrated. The roller round which the rope winds and unwinds has four iron arms, each of which has a leaden poise or ball at the end, just like the end of an ordinary hand dumb-bell. This Knowle example is fixed in an attic and the rope passed through to a gallery beneath. Anyone pulling the rope would get much the same exercise as in pulling a bell rope in a church tower, but without annoying his neighbours by the noise. There used to be a similar apparatus at New College, Oxford.

Figurative sense of "blockhead, stupid person" attested by 1918, American English college slang.

1560s, "pantomime dramatic presentation," from dumb (adj.) + show (n.).

also dumbwaiter, 1749, "a framework with shelves between a kitchen and a dining-room for conveying foods, etc.," from dumb (adj.) + waiter; so called because it serves as a waiter but is silent. As a movable platform for passing dishes, etc., up and down from one room (especially a basement kitchen) to another, from 1847.

"strike dumb with confusion or perplexity," 1650s, dumfound, from dumb (adj.) + ending from confound.

"stricken dumb, confused, perplexed," 1680s, past-participle adjective from dumbfound.

1787, a dialect word in Hampshire, Cornwall, etc. for "a bumblebee." Compare bumble-bee, also dore. The first element likely is imitative (dumble-, bumble-, humble- drumble- all seem to have been used interchangeably).

"inability to speak," Middle English dombenesse, from Old English dumbnes; see dumb (adj.) + -ness. As "stupidity," by 1858.