Can I put my neighbor's sick dog down?
Why do you want their dog to be put down?
Dogs in this condition should be put down to prevent them from having a painful death
I understand that you want to help avoid having this dog go through something painful, but you cannot be sure that he is really suffering. You can only hope that the dog's owners are doing what is best for the dog and its quality of life.
It is fine that you are trying to look out for the dog's best interest, but euthanizing animals is different from how humans would die from diseases. People's quality of life does not decrease to the point where it is inhumane to keep them alive. There is no difference for animals that are suffering, however.
Even if the dog is suffering, it isn't your choice to make. It is your neighbor's dog and they will ultimately make the decision.