Does a warm nose really mean a dog is sick?

Not necessarily. Dog's noses change in temperature quite often due to its surrounding and situations. Of course, the nose will be warm to the touch if your pet just came in from the sun and maybe cold after drinking its cool water. It is best to watch other symptoms of illness like vomiting, diarrhea, weakness, lethargy and change in diet rather than a warm nose. If you are unsure ALWAYS contact your vet for assistance.