It rarely gets the chance to do that, since it is usually either combusted or metabolized (hydrogen-reducing bacteria), to form heavier compounds, along the way.
The other components of the Earth's atmosphere are likewise blown off by the solar wind. The Earth is constantly losing atmosphere to outer space - and has for eons.
Regards,
Since reducing atmospheric temperature is the primary purpose for burning hydrogen, creating water vapor exhaust, the most powerful green house gas, does not seem to fit into the real world equation.
Full disclosure - last chemistry class, almost 60 years ago.
As I think about it, that was almost exactly the same point in time when the French decided to go 100% nuclear, which still seems like a really good idea.