Dog's nose should be moist because wet nose in dogs absorb chemicals present in air and does not allow to inhale them. There is a misconception that wet nose is sign of dog health and dry nose indicates illness. A dog with wet nose can be ill and healthy with dry nose. However, dogs can get cold and have runny nose like humans. If your dog is not lethargic and not showing any sign of illness then you should not worry about his wet nose either sleeping or awake.
