Well you get water from burning hydrocarbons as well. Where do you think the "hydro" in hydrocarbon goes?
Complete combustion of Methane (otherwise known as natural gas) yes I know incomplete combustion is a thing but forget that for a second
CH4 + 2O2 --> CO2 + 2H2O
Produces Carbon Dioxide + Water. Water can dissolve into the air, but in this case it is gaseous. All it does is makes the room more humid.
2H2 + O2 -> 2H2O
Water (gaseous) is produced as well, same as with methane. So, to get your answer, look at what happens when you use your normal gas stove, it's the same thing, except with no soot.
A couple of drawbacks with hydrogen stoves though:
- It's a very leaky gas, only top quality setups have zero hydrogen gas leaks. This exacerbates the next point
- It's highly explosive. Little old Jimmy decides to light a cigarette in his hydrogen + oxygen filled house and poor Jimmy gets blown to kingdom come.