Do heaters dry up the air?

It’s often cold in Seattle, and cold air, of course, tends to make hands dry up. But I notice that turning on the heater can make things even worse, not better. Do air heaters dry up the surrounding air?

They don’t dry up the air so much as reduce the relative humidity. Cold air saturates with water at much lower absolute humidity, so it will take up no more water. When you heat the air the amount of water in it remains the same, but its capacity to absorb water increases greatly, so it will take up water from the surroundings.

Is this a plausible explanation for why turning on the heater tends to be even worse for my hands than cold weather itself?

Yes.

At work we have a dehumidifier in the basement. In the warm dry summers, it collects a gallon of water every day. In the cold damp winter it collects a quarter of that.

So is it a good idea to run a space heater in a bathroom to reduce steam? My bathroom does not have a fan in the ceiling.

For a fixed moisture mass content, the relative humidity of air decreases as temperature increases. So cold air, when heated up, will have low relative humidity.

Cold air, when it gets right next to your skin, warms up somewhat, which is why it seems so dry to your skin. This will also be true if you heat up the air with a space heater or furnace prior to said air making contact with your skin.

@Mahaloth:
A space heater to keep the bathroom warm might help to reduce relative humidity in the bathroom overall, but moisture-related problems/damage will still arise on the window frame, which will still be cold as hell. Lacking a ventilation fan, your best bet for controlling bathroom humidity during/after showers is to leave the door open, particularly after you’re done.

How significant is this difference? Enough to cause condensation in electronics, e.g.?

It sure is. If you bring a laptop in from a cold car you can trash it by starting it up before the condensation has evaporated. You should always wait for it to equilibrate with the temperature inside before firing it up. The same applies to CD drives, DVD players, VCR, pretty much anything.

Ever worn glasses when coming in from the cold? They often fog right up.