Most remodel and cleanout material is fine. The limits come from Connecticut law and from the rules at whichever facility receives the load.
Connecticut makes recycling mandatory for everyone, homes and businesses alike. The state also bans three things outright from disposal: grass clippings, household covered electronic devices such as televisions, monitors, printers, and computers, and lead acid or motor vehicle batteries.
Several other items are handled through state takeback programs rather than the trash, including paint, mattresses, mercury thermostats, gas cylinders, and tires. Plan a separate route for those rather than burying them in a container.
Hazardous waste, liquids, solvents, fuel, and asbestos stay out of a standard container as well. The city code also treats dirt, concrete, and other nonburnable demolition material as a category that needs separate handling, so keep it in its own load.
If you aren't sure where an item lands, ask your account manager before it goes in. A mixed load that gets flagged at the scale can cost you a day.