Almost everything in a Knoxville house, garage, or yard can go on a junk truck — the interesting list is the short one that can't, and the medium one that can with conditions. Here is all three, so nothing surprises anyone on pickup day.
The broad yes

Furniture whole or broken, mattresses, appliances large and small, electronics, exercise equipment, grills, patio sets, carpet and padding, fencing, swing sets, sheds in pieces, yard waste, remodel debris, and the miscellaneous cubic yards every attic produces. Condition doesn't matter for hauling — only for whether it routes to donation instead of disposal.
Yes, with an asterisk
Refrigerators, freezers, and AC units: certified refrigerant recovery first — normal, but it is a line item. Hot tubs and pianos: yes, as disassembly-and-crew jobs quoted from photos. Riding mowers and grills: drained of fuel and disconnected from propane. TVs and monitors: hauled, but routed to e-waste, not landfill.
The hard no list
Liquid paint, solvents, gasoline, oil, and chemicals; asbestos and suspected-asbestos material; medical sharps; ammunition and explosives; and anything radioactive from grandpa's workshop of mysteries. These have their own disposal channels — household-hazardous-waste events for most — and no legitimate hauler bends on them.
When in doubt, photograph it
The item you are unsure about has been photographed at booking a thousand times before: the freezer with contents, the piano nobody plays, the shed that leans. A photo gets a yes, a no, or a price in minutes — and it means pickup day contains zero surprises, which is the entire goal.
The rule behind the lists
If it is solid, non-hazardous, and physically leaves the building, it hauls. If it pours, fumes, or requires a license to touch, it has its own channel — and the county's hazardous-waste events are the free answer for most of that shelf.
Photograph the edge cases at booking, and pickup day stays boring — which, in this business, is five-star service.