Unless there's a water pipe in a wall that might freeze I wouldnt think cold would hurt it. But then again, I never have lived in freezing territory.

What might concern me is humidity. Too humid and mold grows on things.

I have a section of the house that's closed off and have watched at times the humidity slowly creap upwards. When too humid I just open it up to the climate controlled main house during the days for two or so days. When the weather is overly humid, I just put a dehumidifier back there and leave it closed off.(except then I gotta go dump the bucket for a while)

As mikie says, it's not the heat, it's the humidity!

In cold climates, it's the freeze and thaw that does in houses - everything moves, just as roads do when they make pot holes in the spring. After a while the expansion and contraction from the temperature change causes stuff to move: plaster to crack and fall off the wall, windows and doors to shift, pipes to disconnect, etc.
We close off parts of our house, especially our guest room with electric heat! As it is on the first floor I use it for cold storage for food, so I open the door every day or so. We just closed it up this month, including winterizing the toilet. By late March/April (mud season here) we will be ready to open the room again. So it is not subject to much freeze and thaw.
My parents closed off much of their house in New Hampshire. They lived in 13 rooms in the summer, not including the 5 porches!, but only 4 in the winter, the kitchen, the library (my mother's office) my father's office, and their bedroom. They had no problems at all. When company came in winter it took a day to heat the cold rooms though! We had lots of wool blankets too.

If it gets too cold lead paint peels and falls off (I.e. your first layers of paint on walls and trim). HOwever it'll probably get enough warmth from the adjoining rooms that that won't happen. Probably wouldn't get any colder than your basement say (~ 50 degrees?) I'd try it and just see how cold it gets in there.