Everyday remodel and cleanout debris is fine. The limits here come from the receiving site and from a few firm city rules.
The city's own refusal list is long and specific. It turns away all waste from commercial operations, heavy construction and demolition rubble, hazardous waste, medical waste, special waste such as contaminated soil, pesticide containers, sandblasting waste, and sludge, materials containing friable asbestos, radioactive waste, railroad ties and creosote-treated material, animal carcasses, boats, commercial and industrial tires, and oil drums unless both ends are cut out and rinsed. Engine blocks, transmissions, and axles only go in if they've been disassembled and drained.
Hazardous waste, liquids, wet paint, solvents, and fuel never belong in a container. Lithium-ion, rechargeable, and nickel-cadmium batteries can't go in the trash at all and have to be taken to the city's Resource Recovery Center. Electronics are accepted there too, with the exception of anything containing a cathode-ray tube.
City rules also bar placing any container or receptacle on, in, or over a storm drain, which rules out some tempting curbside spots.
If you're unsure about something, ask your account manager before it goes in the container.