Buoyancy describes a hot-air balloon’s rise caused by warmer air inside its envelope.
When air inside the envelope is heated, it expands and becomes less dense than the cooler air around the balloon. The surrounding atmosphere then provides an upward buoyant force. If that upward force exceeds the combined weight of the envelope, basket, burner, fuel and passengers, the balloon rises.
The balloon does not rise because the burner creates thrust like an airplane engine. The burner only changes the temperature and density of the air inside the envelope. Pilots therefore control altitude by heating or allowing the air to cool.
Buoyancy also explains why adding weight makes a balloon descend and why releasing ballast is not the normal control method for a modern hot-air balloon.