Yes dogs do feel pain when they give birth to the puppies.
The dogs puppies are smaller than a human baby so it's likely that the dog experiences less pain or at least less intense pain than a human giving birth to a bigger baby.
Dogs never seem to holler or whimper though when they give birth which is amazing as I was screaming when I gave birth because it was really painful for me to give birth.
But the pain was so worth it once I held that baby and still worth it when I'm around my children now.
Dogs also go into labor much sooner than humans do and they keep their pain more private for some reason but they may show other signs of distress.