Nope. Zombies are dead, and are oblivious to both heat and cold. They assume the temperature around them, making them in effect a cold blooded creature.
But to my knowledge, this has never been addressed because until a very certain point, zombies really don't do anything beyond killing and eating. Romero in his later films- Day, City, and Land of the Dead Played with the idea of zombies developing intelligence. However, we have no evidence that zombies perform higher functions like sweating considering they don't even perform lower functions like breathing. The only time we ever see a zombie even slightly moist is when they have been wetted by an outside agency- rain, river, etc.
Zombie thermoregulation has been addressed in the sense that it is generally shown that the assume the temperature of their surroundings. That either makes them cold blooded or a Black Box, I suppose.
Edit: I would also add that sweating assumes a digestion-based intake or water and salts and such. However, any time the topic of zombie digestion comes up, it is always that hey don't digest their food, it simply rots in their stomachs. They do not actually require eating
Some variants, however: Return of the Living Dead, zombies are intelligent but also show no signs of sweating. Any and all biological tissue can be reanimated, but with the sole purpose of consumption of brains. I suppose that isolated sweat glands could be deactivated, and if put into a solution, could pump it through. Knowing those movie, they would hyper-contract to blind a human in order to...get at their brains somehow?
Also, with modern diseased zombies, that is humans who have been infected with a virus making them zombie-like and typically hyper aggressive and fast, there absolutely is the potential for sweat since they are still alive and require things like food, shelter, heat in order to survive-their minds are gone is all. In fact, with that level of aggression and such, I would imagine that they have a somewhat elevated temperature and would be sweating nonstop.