It's only green potato skins that are toxic to humans. It seems that the green skins and sprouts are also poisonous to dogs, but other than that they should be fine.
Potato poisonings among people and dogs have occurred. Solanum alkaloids can be found in green sprouts and green potato skins, which occurs when the tubers are exposed to sunlight during growth or after harvest. The relatively rare occurrence of actual poisoning is due to several factors: solanine is poorly absorbed; it is mostly hydrolyzed into less toxic solanidinel; and the metabolites are quickly eliminated. Note that cooked, mashed potatoes are fine for dogs, actually quite nutritious and digestible.