Hydrogen is flammable, and for any fire to burn it needs oxygen. Why does a compound made of hydrogen and oxygen put out fires instead of catalyzing them? I understand that hydrogen and water are chemically different compounds, but what causes water to be non-flammable?
You can think of water as the ash from burning hydrogen: it's already given off as much energy as possible from reacting hydrogen with oxygen.

You can, however, still burn it.  You just need an even stronger oxidizer than oxygen.  There aren't many of them, but fluorine will work, 

$$
\ce{2F2 + 2H2O -> 4HF + O2}
$$

as will chlorine trifluoride:

$$
\ce{ClF3 + 2H2O -> 3HF + HCl + O2}
$$
