I have not had a dog around my rabbits much yet since they are in cages in a small barn- not much room for a dog right now! However, I have had two herding dog mixes around our poultry for over 12 years, as well as a two retrievers.
Herder 1 started out a bit crazy and we had to teach him how to behave around the chickens. He went from going after the chickens if they flew to being totally trustworthy and helpful around them. They learned very well to trust him, and never minded him going around and through their midst. He was a calm, controlled, precise type of dog, unless a squirrel got the best of him.
He even learned to bark at birds of prey in the sky so we would be aware of them and could put the chickens away or watch them for safety! He really enjoyed that job and did well with it.
We then got Retriever 1 that had little training and was pretty energetic and fast-moving. We had her for part of the same time as Herder 1. She never bothered the chickens, but the chickens never seemed to trust her the same as Herder 1. She seemed too quick-moving for their liking, although they were never too upset by her. They just totally relaxed around Herder 1, but wanted to keep an eye on Retriever 1.
Next was retriever 2, a puppy. We worked with her to help her learn to be with the chickens and not chase them. That was actually pretty easy for her to learn. Then she would just hang out, exploring their yard and playing with the ducks now and then. The chickens didn't really mind her much, but didn't relax quite as much as with Herder 1.
Next was Herder 2. He just fit right in and had no issues with the chickens or turkeys. This was a new flock that had no experience with dogs, too. I expected him to be harder to have around the flock, but he is great. He has the same smooth, controlled behavior that the chickens liked about Herder 1. He is high energy, but calm and not an issue at all around the birds.
I have always thought it so interesting how animals like this learn to be around each other and trust each other. Farmers have had dogs herding and protecting all kinds of animals and I think certain breeds/dispositions seem to cause the animals they are around to relax more than others naturally. It's a good thing for a farmer and his animals to have a good dog around!