Most puppy panting is NORMAL, particularly when they are sleeping. They are still growing and they can have some pretty active dream states. It doesn't look normal, and it can scare the pants off of you, but in most cases it’s nothing, especially if there are no other symptoms, such as coughing, lethargy during the day, vomiting, or lack of appetite.
Puppies pant, particularly when they sleep. Dogs don’t cool themselves evaporatively through their skin like we do, they release bodily heat by panting (they do have a few sweat glands on the bottoms of their feet, but that’s it).
Some breeds have a thick coat, so they can get hot, which contributes to panting, so if you’d like you can try reducing the temperature or placing a fan on your puppy.
There are times that panting is a sign of something serious, so if you are in doubt about your puppy panting being normal, please call your vet.
Here’s an informative link about puppy panting: http://www.dog-health-handbook.com/panting-puppy.html