No you won’t, all the huts we will be staying in are heated and staffed and there will be either blankets or duvets provided. You are actually more likely to be too hot at night than too cold. A lightweight sleeping bag liner (silk or cotton, see equipment page for details) is an option however.
Officially everyone staying in a mountain hut is supposed to have one of these, but the rule is seldom enforced. If you would like one for hygiene reasons by all means bring one, but if in doubt it’s probably better to do without, as it is just one more thing to carry (most of the guides don’t bother).
Dormitory in the Vittorio Emmanuelle Hut