What will happen if you put old gasoline in a lawn mower?

DON'T DO IT !! Although there will be no physical damage, the old gasoline is likely to gum up the carburetor, fuel lines, and the spark plug - not to mention the walls of the carburetor itself and the head cylinder. You will have trouble starting the engine and will have to remove the carb and components to clean everything off - NOT WORTH SAVING A BUCK OR THREE to keep from buying new gasoline.