Here I will cover the teeth and skin around the mouth, and tongue.
Rabbits are often born with this. It is in their genetics. When the bottom set of teeth come up over the top, and split and eventually break off, this is a malocclusion. Breeders cull these rabbits since it is passed on to the next litter. This doesn't become life threatening unless the teeth make it where the rabbit can't eat. Some rabbits even have mild malocclusions where the teeth don't even grow long or split. Teeth should be clipped with nail clippers, or if you're a pet owner, take it to a professional. The teeth clip rather easier then you would think, but it is often hard to get the rabbit to sit still while you chop off it's teeth. There are rumors that this comes from inbreeding, but since I have bred mother to son, etc, I haven't seen it alot in my herd. I have never bred blood brother to blood sister before. Perhaps this is where it comes from, somewhere in the line?
This is often from an injury from fighting, or from chewing on the cage or something else. This could sometimes lead to a Malocclusion. Teeth will probably need to be clipped, and the rabbit should be taken to a vet to be checked out.
This is where the rabbit seems to be drooling and fur is matted and wet around the mouth and neck. This is often from an abcessed tooth. If the dewlap is what is wet, then this is probably Dermatitis. This is often with an older doe with a large dewlap (skin below the mouth on the neck) and it drags on the cage floor and gets into the water dish. The best thing to do, would be give her a water bottle instead. If you don't do this, she could end up getting infested with maggots and end up dying. The best thing is prevention. If however it is Slobbers, you must see a vet to get the tooth either clipped or surgically removed.
This is from very poor sanitation. They are often found around the mouth and neck. It happens when the rabbit gets a cut or a score, bacteria invade the body and this sets in. These rabbits will die without treatment, but it is best to cull the animal and clean all your equiptment.
The tongue rarely has anything to do with disease, other than mouth cancer, perhaps. When the tongue is blue, it is a sign of pneumonia. When doing an autopsy, soon after death, if the tongue is allready blue, then it was probably snuffles or pneumonia that killed the rabbit.