If the rat that the dog bit or ate has a disease then yes the dog can catch that disease from the rat they bitten or have eaten.
If your dog tries to eat a rat you should try to take it away from them and if you notice any dead rats or dead mice around your yard or house remove them immediately and throw them in the trash can.
If your dog does bite a rat or eat a rat just keep an eye on them to see if the get sick and if they do start vomiting or have a loss of appetite etc then you should take the dog to the vet if possible to be checked out.
Rarely does a dog get sick from biting a rat or eating a rat but they can if the rat has been diseased.
Also if the rat got into some rat poison and died and the dog ate the dead poisoned rat then the dog could be poisoned as well.