burden (n.1) Look up burden at Dictionary.com
"a load," Old English byrðen "a load, weight, charge, duty;" also "a child;" from Proto-Germanic *burthinjo- "that which is borne" (cognates: Old Norse byrðr, Old Saxon burthinnia, German bürde, Gothic baurþei), from PIE root *bher- (1) "to bear, to carry; give birth" (see infer).

The shift from -th- to -d- took place beginning 12c. (compare murder (n.)). Archaic burthen is occasionally retained for the specific sense of "capacity of a ship." Burden of proof is recorded from 1590s.
burden (n.2) Look up burden at Dictionary.com
"leading idea," 1640s, a figurative use from earlier sense "refrain or chorus of a song," 1590s, originally "bass accompaniment to music" (late 14c.), from Old French bordon "bumble-bee, drone," or directly from Medieval Latin burdonom "drone, drone bass" (source of French bourdon, Spanish bordon, Portuguese bordão, Italian bordone), of echoic origin.