Milk and dairy are not toxic for dogs. However, dogs can be lactose intolerant and some may even be allergic to milk protein. Even in dogs that can digest lactose, large amounts of milk can cause intestinal distress and diarrhea.
Unless aware of a dog's ability to process lactose, it is best to give it milk in very low quantities. Milk, whole and skim, contain 11 grams of lactose per cup. That is enough to make a dog sick if it is lactose intolerant.
Most other dairy products contain lactose in lower quantities. A spoonful of ice cream only contains about 2 grams of lactose, making it a good way to test a dog's tolerance.