What force makes a hot-air balloon rise when its air is heated?

Buoyancy makes a hot-air balloon rise when the air inside it is heated.

Heating the air does not make the balloon rise because flames push it upward. Instead, warmer air expands and becomes less dense than the cooler air surrounding the balloon. The surrounding atmosphere then provides an upward buoyant force, just as water supports a floating boat.

A balloon rises when the upward buoyant force exceeds the combined weight of its envelope, basket, burners, fuel, and passengers. To descend, the pilot can stop heating the air or open a valve at the top so hot air escapes. The balloon then becomes heavier relative to the air around it.

This explanation is often confused with the behavior of helium balloons. Helium balloons also rise through buoyancy, but their lift comes from helium being less dense than air rather than from heated air. A hot-air balloon carries its lifting gas inside a fabric envelope and must continuously manage temperature during flight.