"a load, that which is borne or carried," Old English byrðen "a load, weight, charge, duty;" also "a child;" from Proto-Germanic *burthinjo- "that which is borne" (source also of Old Norse byrðr, Old Saxon burthinnia, German bürde, Gothic baurþei), from PIE root *bher- (1) "to carry," also "to bear children."
The shift from -th- to -d- began early 12c. (compare murder (n.), rudder, afford). Archaic burthen is occasionally retained for the specific sense of "capacity of a ship." Beast of burden is from 1740. Burden of proof (Latin onus probandi) "obligation on one party in an action to establish an alleged fact by proof" is recorded from 1590s.
"leading idea, main topic," 1640s, a figurative use (on the notion of "subject often repeated") of the earlier sense "refrain or chorus of a song," 1590s, originally "bass accompaniment to music" (late 14c.), from Old French bordon (Modern French bourdon) "bumble-bee, drone," or directly from Medieval Latin burdonom "drone, drone bass" (source also of Spanish bordon, Portuguese bordão, Italian bordone), of echoic origin.