Natural sausage casings are made from intestines. There is nothing wrong with eating the intestines themselves. You can search online to find some methods to clean them for human consumption to see if you might want to try one of those methods for cleaning them.
I know your concern is what is in the intestines, not the intestines themselves. If you withheld food before you butchered (many people do) there won't be much in them to start with. But there will be a bit.
The way I look at it, if your dog catches and eats a rabbit, rat, or anything else it will usually eat the intestines. Many predators eat intestines. It is disgusting to you and me but they are filled with highly nutritious partially digested food and animals are not going to pass up a good source of nutrition. I'm not sure how much nutrition is in the intestine walls themselves so I won't speculate on that.
Most diseases are species specific. It is highly unlikely they will be carrying any disease that will infect your dog. We feed internal organs to dogs all the time without a worry about infections. It's possible there could be a parasite like roundworms that could be passed on. Did you look in the intestines for tapeworms or roundworms when you were butchering? I do, if the one I butchered has them the rest of the flock does too. I've never seen either one in the intestines of my chickens. If your dog and chickens share the same territory if one has them the other probably does too.
I haven't seen one recently but there have been a few threads on the forum about dogs eating chicken poop. Nobody reported any ill effects from that, it's just disgusting to many people. I shut one of those threads down when I commented that I don't worry about dogs eating poop but I don't French kiss my dogs so I don't have to worry about where their tongue has been.
Personally I don't feed the intestines to my dogs. I cut them into 2" lengths and feed them to the remaining chickens without cleaning them.