Most remodel and cleanout debris is fine. The limits here come from New Hampshire law and from what the city landfill will take.
New Hampshire now bars rechargeable lithium-ion batteries from landfills and incinerators, and the city states plainly that households and businesses must recycle them instead and that putting them in the trash is illegal. That reaches cordless tool packs, e-bike and scooter batteries, lawn equipment, phones, laptops, and tablets, which is a long list on a renovation site.
The Four Hills Landfill turns away ammunition and fireworks, antifreeze, battery acid, boats, cleaners, gasoline, large tree stumps, medical and infectious waste, oil-based paints and stains, pesticides, pool chemicals, sharps, and used motor oil. City law separately bars hazardous, toxic, radiological, and incompatible waste from the landfill, and a violation can cost a hauler its permit.
Asbestos is a special case. The landfill does take it, but only from Nashua sites, only with proper notice and compliance, and only when it has been pre-approved in advance. Never put it in a container without clearing it first.
The city also keeps rocks, concrete, asphalt, and dirt out of its automated collection carts, and reserves the right to impose size limits, quantity limits, preparation requirements, and documentation on the source of a load. If an item has you unsure, ask your account manager before it goes in.