No. A rocket does not need to fire its engines to stay in orbit. It does, however need fuel and oxygen to perform any maneuvers, so a rocket that runs out of oxygen will be stranded in space.
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The first stage is jettisoned, to fall back to earth, as the fuel runs out.
The oxygen tank is needed for the rocket engine to burn fuel. The engine requires both fuel and oxygen to create combustion for propulsion. Without oxygen, the fuel cannot burn and the rocket will not be able to generate thrust.
Astronauts on the moon need to bring their own oxygen supply because there is no breathable atmosphere on the moon. Without a source of oxygen, they won't be able to breathe and survive on the lunar surface.
The solid rocket boosters are jettisoned as their fuel runs out. Their fall is slowed by parachutes and they are later recovered and returned to the Kennedy Space Center for refurbishing and reuse on later missions.
Spaceships run on various sources of energy, such as rocket fuel like liquid hydrogen and liquid oxygen for chemical rockets, or nuclear power systems like radioisotope thermoelectric generators (RTGs) or nuclear reactors for deep space missions. Solar panels can also be used to generate electricity for powering spacecraft systems.