Old English beatan "inflict blows on, strike repeatedly, thrash" (class VII strong verb; past tense beot, past participle beaten), from Proto-Germanic *bautan (source also of Old Norse bauta, Old High German bozan "to beat"), from PIE root *bhau- "to strike."
Past tense beat is from c. 1500, probably not from Old English but a shortening of Middle English beted. Of the heart, c. 1200, from notion of it striking against the breast. Meaning "to overcome in a contest" is from 1610s (hence the sense of "legally avoid, escape" in beat the charges, etc., attested from c. 1920 in underworld slang). Meaning "be too difficult for" intellectually or physically (by 1870) is behind the shrug-phrase beats me.
Meaning "strike cover to rouse or drive game" (c. 1400) is source of beat around (or about) the bush (1570s), the metaphoric sense of which has shifted from "make preliminary motions" to "avoid, evade." Nautical sense of "make progress against the wind by means of alternate tacks" is from 1670s. Command beat it "go away" first recorded 1906 (though "action of feet upon the ground" was a sense of Old English betan); it is attested in 1903 as newsboy slang for "travel without paying by riding on the outside of a train."