I have a wood stove in a vacation home -- when the stove is not being used for a few weeks at a time should I keep water in the cast iron steamer pot or empty it?
I can't see any harm one way or the other. The steamer will only help the humidity when the stove is running, as the heat from the stove is what drives evaporation.
Personally we don't use one any more, simply because we don't mind the feeling of the slightly drier air. It is fine once you get used to it, but this varies depending on people's skin. People with naturally dry or weaker skin might fell less comfortable.
We don't usually notice much difference between having the central heating running or the fire lit either - the air seems equally dry - but with the steamer on the fire we can actually do something about it.