Assuming that it it is the thermostat that's stuck, I'm guessing that it's stuck open, which is why your heater won't warm up.
If it was stuck closed, your engine would overheat, as the water wouldn't be able to circulate and take away the heat from the engine and the heater would heat up as well.
Either way, it's not advisable to drive with it sticking ......
Just thinking about it, it would prob not do any harm to your engine, driving whilst the thermostat is stuck open. It's other issues you'd have to contend with, for example your heater would take forever before it started blowing warm air, which could cause problems in this weather .......
If it was stuck closed, I would def advise against driving, as you'd prob end up causing the engine to overheat and causing all kinds of problems ......