First let's look at the most innocuous reasons your dog is lethargic and not eating. It could be that it's a hot day out and your dog is tired. If he doesn't want to go for his daily walk, it could be that his muscles are sore from yesterday's long hike. Your dog may even not have much of an appetite due to the weather, but if a day or two goes by and the dog still seems lethargic and without appetite, it's worth giving the vet a call, as this could be a sign of illness. In addition, you might want to check your pet's weight. A gain or loss of 10 percent or more is something that the vet should be aware of. If your dog has always been a gentleman of a dog, and all of a sudden he starts going through the garbage or taking food that's supposed to be for the family's meal, he's asking for a check-up at the vet.
Similar to people, if a dog starts drinking much more than usual, this could be a symptom of diabetes. Even if you don't notice how much the dog drinks, but if it starts having accidents in the house or scratching to go outside all the time, these could be signs of increased water intake.
Yet other symptoms of illness in dogs are frequent vomiting, fever, bloody diarrhea, increased drooling, wriggling worms in the dog's feces, red or swollen gums, trouble breathing, eyes that are red, cloudy, or with a discharge, an inability to use its back legs, whimpering when touched, fainting or convulsions, or an enlarged abdomen or lump. Be in contact with the vet instead of simply worrying when you notice any of these symptoms.