Burning is the more common name for the Chemical Reaction of Combustion (and one type of system in which combustion takes place is a Fire ) though the term burning may also be used when referring to a self sustaining Thermonuclear (nuclear fusion) reaction.
A Burning At The Stake is an execution by fire, usually by being burnt to death on a Pyre . Immolation is destruction by fire, although it often refers specifically to Suicide by burning.
A burn is the Firing of a Rocket propulsion system.