I'm assuming that you are talking about pre-mixed burners. If you have no pre-mixing of the gas with oxygen/air, the answer to your question is trivial: No combustion without both burnable gas and oxygen. However, all good burners use pre-mixing: The gas is mixed with air in a separate mixer before it enters the actual burner. And for those, the answer is no that trivial.
The trick of pre-mixed burners is, that every combustible gas mixture has a certain flame speed. If the gas mixture moves faster than this flame speed, the flame front is basically blown away with the mixture.
Now, when you put a combustible gas mixture through a tube at a speed that exceeds its flame speed, and let it exit this tube, tiny vortexes will be created at the edge of the gas stream and the gas at the edges will be slowed down by the surrounding air. This slowdown in combination with recirculation of hot gases by the turbulences allows the flame front to attach to the edge of the tube, and then eat its way progressively into the unburned mixture flow. In other words, the edge of the tube acts as a flame holder.
Flame holders may be constructed in different ways: They may be simple wire meshes (some stoves), or they may be tube edges (typical cooking stoves), or they may be more elaborate like vortex combustion chambers (blow-torches). But they all operate on the same principle that the flow-rate of the pre-mixed gas is faster than its flame speed, making it impossible for the flame to enter the openings of the gas supply backwards.
The wikipedia article on pre-mixed flames seems to have more details on this.
(Btw, that's why every gas stove has a certain minimum flame size: If you regulate a gas stove down too much, the speed of the mixture becomes lower than the flame speed at the flame holder, and a flash-back occurs. Stove designers don't want this for obvious reasons, so they construct the valves to avoid this condition.)