The natural materials in a traditional gun case are hydrophilic, ie. they absorb moisture. However they are always happy to share their burden with your gun and so rust will occur under most conditions. Only exception is under conditions of very low humidity where there simply is no water to share!
I learnt this lesson the hard way, leaving my first new bought gun in a case over night before an early start. On arrival at the venue, I took out the gun to find the barrels had turned a delightful, light shade of rust. No long term damage done but I have never left a gun in its case for a moment longer than absolutely necessary since.
