If you are dog-tired, you are extremely tired — so tired you want to drop. It is informal and warm, the sort of thing you say after a long day of work or walking, not a medical word and not very formal. It sits with phrases like dead tired or worn out. People have long said someone is tired as a dog; this adjective packs that picture into one word. Why a dog? A dog that has run and worked all day really does flop down — that is the likely story. It describes how you feel, not how you look, and it does not mean you are tired of dogs.

累慘了、筋疲力盡（口語）