Why is it necessary to burn the hydrogen fuel coming out of the engine for the lift of the rockets?

If it is done to create a greater reaction force on the rocket then why can't we do the same lift with just adjusting the speed of the hydrogen gas going out of the engine like we can release them at a great pressure (and also by adjusting the size of the nozzle opening) and thus at a greater speed?

Is it possible for rockets to fly without burning the fuel and also without affecting the volume of the total fuel i.e. with the same amount of fuel used in general propellers? (I know they are too massive but consider a lighter one.)