I agree, it depends on the disease. I would not eat any rabbit with an infection in their blood or meat at all. (Like an infected wound.) But a rabbit culled for coughing I would eat.
I kind of handle it like hunting. If I could not tell the animal was sick after I had shot it on a hunting trip, then I'm fine with eating it from my herd. But if I can look at the animal's carcass and tell it had an issue, then I would not eat it.
Different people have different opinions on this one. Technically the heat of cooking would kill any virus or bacteria. But no one is sure what changes have happened to the meat as far as nutritional value goes. And there are diseases like scrapie in sheep (CJD or Mad Cow in cows) that are not killed by heat. So do your own research and make your own choice on this issue, because no one has done full research on rabbits to truly know the possibilities for disease transfer that may or may not exist.
If the internal organs are all healthy and there are no sores on the skin, then it should be fine. But if the lungs are a weird color, liver has spots or other such things, I wouldn't eat it. Would give it to the dogs instead at that point.