"burning" is a violent reaction between a substance and oxygen (either in elemental form, O2, or taken from some other compound, such as saltpeter, KNO3). Now, burning is, by definition, a red-ox reaction, and the laws of redox tell us that an atom cannot have an oxidation state that is higher than the number of electrons it can lose to obtain a full octet, or lower than the number of e- it can gain to have a full octet. Since hydrogen can only gain or lose one electron, its max oxidation state is +1, so in H2O hydrogen can no longer be oxidized. If you don't know what red-ox is (I didn't learn it in my first year of high school chemistry), then I could explain it in more depth.
Of course, many reactions *look* like a combustion reaction, such as that of flourine gas with, say, iron (or pretty much anything, including water), where the flourine acts just like oxygen would, and it looks as if the iron is burning, but in the technical sense, it isn't really, since no oxygen is involved:
2Fe (III) + 3F2 ----> FeF3 + lots of heat
Also, many reactions (such as iron rusting) would seem to be a combustion reaction; after all, the reaction is the same, whether you are burning iron or just letting it rust. (many people don't realize it, but iron burns very easily: just put a match to some steel wool. Not very exciting, but the iron's definitely burning).
Now I could be entirely wrong with my definition of "burn", but the red-ox bit is the reason for water not burning.