"timber sawn into rough planks for use," 1660s, American English (Massachusetts), earlier "disused bit of furniture; heavy, useless objects" (1550s), of uncertain origin. It is said to be probably from lumber (v.1) on the notion of "awkward to move," and perhaps to have been influenced by or associated with Lombard (q.v.), the Italian immigrant class famous as pawnbrokers (and money-lenders) in old England. Lumbar and Lumbard were old alternative forms of Lombard in English.
The evolution of sense then would be because a lumber-house ("pawn shop; place where thieves stash stolen property") naturally accumulates odds and ends of furniture. The 19th century guess was that it comes directly from lumber-house or lumber-room in the pawn shop sense, but these are not attested before lumber (n.). Lumber camp is from 1841; lumber-mill is from 1830; lumber-yard is from 1777.
Live Lumber, soldiers or passengers on board a ship are so called by the sailors. [Grose, "Dictionary of the Vulgar Tongue," 1785]