Water being non-flammable is almost exactly the reason why hydrogen and oxygen are so flammable.
Water is incredibly stable; it has a very low amount of potential energy. On the other hand, hydrogen and oxygen separately (in diatomic form) have a high amount of potential energy; they are not so stable, and ideally want to dump that potential energy into heat.
So what happens? The hydrogen and oxygen react, releasing all of that potential energy as heat. They burn. And what do you get? Water, with very little energy left over to do anything.
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