Short answer
Liquid paint usually can't go in a dumpster. Completely dried-out latex paint cans are often accepted; oil-based paint goes to hazardous waste drop-off.
Latex vs oil-based
Latex (water-based) paint can often be dried out (leave the lid off or mix in cat litter) and put in the trash; Boston, for example, tells residents to dry latex paint and only bring oil-based paint to its drop-off days (City of Boston: Zero Waste Day (latex paint)). Rules vary, so check your city.
Oil-based paint, stains and solvents count as household hazardous waste. Take them to your local household hazardous waste site or event: every city guide lists one.
PaintCare drop-off
In states with paint stewardship laws (including California, Colorado, Connecticut, DC, Maine, Minnesota, New York, Oregon, Rhode Island, Vermont and Washington) PaintCare runs free drop-off sites, mostly at paint and hardware stores (PaintCare: paint drop-off program).
Local drop-off sites
Each of our city guides lists the official household hazardous waste and drop-off option: New YorkLos AngelesChicagoHoustonPhiladelphiaSan AntonioSan DiegoAustinSeattleSan FranciscoDenverWashingtonBostonMinneapolisPortlandNashvillePittsburghAtlantaColumbusIndianapolisCharlotteTampaKansas CitySt. LouisDetroitMilwaukeeSacramentoCincinnatiChattanooga.
Questions
Can I put empty paint cans in a dumpster?
Empty, dry cans usually yes. Cans with liquid paint usually no: dry latex paint out first, and take oil-based paint to hazardous waste drop-off.