It is well known that to give a liftoff a rocket, we use the momentum principle: The momentum of the gases emitted by a rocket and the rocket's momentum is equal to zero. So why is it not enough to emit any gas quickly to impart a rocket's velocity without burning it? What is the contribution of burned gases to the rocket's momentum?
Combustion isn't strictly necessary, and there are kinds of rocket engines that don't rely on combustion. But 'emitting a gas quickly' enough to lift a 10,000 kg rocket into space is not a trivial affair -- somehow you have to speed that gas up enough to achieve the required thrust on the rocket.
That means releasing energy into the gas, and a lot of it. Combustion is a good way to do that -- explosions release a lot of energy. There are other ways, but most of them can't produce the kind of power you need to get to space.
So basically, it is enough to emit a gas quickly. Combustion is just how we make that happen.
Well you know that a rocket works using Newton's third law. Propellant goes one way, and the reactionary force propels the rocket in the opposite direction. This means that anything that is able to direct a force that is greater than that of the rocket's weight at any given moment is able to propel a rocket off the ground. Including any type of gas that may or may not be a product of combustion. For example, you can make a simple, yet effective, hobby rocket by pouring liquid nitrogen into a bottle of water.
Hydrogen -- a light and extremely powerful rocket propellant -- has the lowest molecular weight of any known substance and burns with extreme intensity (5,500°F). In combination with an oxidizer such as liquid oxygen, liquid hydrogen yields the highest specific impulse, or efficiency in relation to the amount of propellant consumed, of any known rocket propellant.
Combustion is most preferred as this means more efficiency and higher thrust per weight unit.
A type of rocket thruster is a cold gas thruster. This is just a pressurized gas that is released from a small nozzle, at high velocity. Cold gas thrusters could theoretically provide the thrust that's required for sustained flight, but is not used due to efficiency concerns .
As opposed to traditional rocket engines, a cold gas thruster does not house any combustion and therefore has lower thrust and efficiency compared to conventional monopropellant and bipropellant rocket engines. Problem is, that compressing gases take up a lot of energy. And there's also solid fuel sources. NASA uses solid rocket boosters as they have a lower specific impulse. Which means it burns highly energetically in a very short amount of time.
In a nutshell, combustion is used as opposed to any other form of propellant is because this allows for more thrust per kilogram of fuel. As for the part about "contribution of burning gas to the rocket's momentum", there is no such thing. It's all about efficiency.