The large houses of the well-heeled of Britain commonly had much very old, well-built furniture in them, more than was to be used in every room at any given time. Every piece was made-to-order. When not needed, it was neither sold nor discarded by them. At least one out-of-the-way room was selected to store the pieces that were not in use. This was called the lumber room. Such is what is alluded to in the definition in the Oxford English Dictionary (OED), whose first reference is Richardson's novel Pamela and which is mentioned in a bit more detail in Daniel Pool's literary reference book of the 1990s, "What Jane Austen Ate and Charles Dickens Knew," among other literary reference works.
The phrase "lumber room" is found in British fiction at least during the 19th century (e.g., Arthur Conan Doyle's 1891 Sherlock Holmes short story "The Five Orange Pips"), and the use of the word lumber in this phrase is that found in many obsolescent turns of phrase heard in various English-speaking countries. Probably one of the most evocative references is the short story by "Saki" (H. H. Munro) called "The Lumber Room": "Often and often Nicholas had pictured to himself what the lumber-room might be like, that region that was so carefully sealed from youthful eyes and concerning which no questions were ever answered. It came up to his expectations. In the first place it was large and dimly lit, one high window opening on to the forbidden garden being its only source of illumination. In the second place it was a storehouse of unimagined treasures."
The OED mentions in the verb "lumbering" that it first meant to obstruct with pieces of wood to make things from, and then shifted to general obstruction, hence furniture fit the later meaning.