According to the ASPCA, puppies should not drink cow's milk, as it can cause digestive problems. For the first month of their lives, puppies can subsist on milk from their mothers.
WebMD.com advises against feeding dairy products to dogs in general, as they can lead to digestive issues and allergies. Some canine digestive systems cannot properly digest lactose; lactose intolerance in dogs manifests as diarrhea, vomiting and symptoms that can be confused with allergies. While puppies four weeks or younger get all necessary nutrition from their mothers' milk, if that milk is not available, there are commercial alternatives with which to replace it.