I’m glad he seems ok. A Dachshund can kill a rabbit in an instant, in fact that’s partly how Dachshunds are used, so I’m guessing your dog was more about playing than acytually hunting. That’s great. One good thing about rabbit’s fur is that it’s very dense and it comes off when the bun gets scared, and that offers some protection against bites.
You will of course need to check him over. Remove the dog so he doesn’t try to join in. Don’t punish the dog obviously. Try to make the room quiet and everything really calm.
Look for sores and scratches, gently feel his legs and feet. Look at his eyes and ears. If he accepts being sat in your lap that’d be good. Be two people when you do it, one that holds the bun and one that does the palpation (feeling). If he lets you, wrap him in a fleece blankie and warm him up with your own body heat.
If he has suffered grave internal injury, I’m sorry to say that a vet probably can do very little for him.
Will he eat? Will he take a treat? Fear can make a bunny loose its appetite and even lead to gastrointestinal stasis, a very serious condition where the intestines aren’t moving so food can’t pass along their lenght. A bunny tummy must never get empty.
Please report back as you go along.