c. 1200, lode, lade "that which is laid upon a person or beast, burden," a sense extension from Old English lad "a way, a course, a carrying; a street, watercourse; maintenance, support," from Proto-Germanic *laitho (source also of Old High German leita, German leite, Old Norse leið "way, road, course"), from PIE root *leit- (2) "to go forth" (see lead (v.1)).
It seems to have expanded its range of senses in early Middle English, supplanting words based on lade (v.), to which it is not etymologically connected. The older senses went with the spelling lode (q.v.). The spelling is modern. Meaning "amount customarily loaded at one time" is from c. 1300; meaning "a quantity of strong drink taken" is from 1590s. Meaning "the charge of a firearm" is from 1690s.
Meaning "a great amount or number" (often loads) is from c.1600. Figurative sense of "burden weighing on the mind, heart, or soul" is first attested 1590s. Meaning "amount (of work, etc.) to be done by one person" is attested in compounds from 1939 (first was workload). Colloquial loads "lots, heaps" is attested from c. 1600. Phrase take a load off (one's) feet "sit down, relax" is from 1914, American English. Get a load of "take a look at" is American English colloquial, attested from 1929.