Can an electric stovetop burner be damaged from prolonged use without cookware on it?
An older stove element likely uses no active thermal regulator. It will run hotter, brighter when you remove cookware. It should be safe to run this way (from an electrical point-of-view).
However, watch for a hot spot, where one small section of the element is brighter...an indication that the element is damaged, at risk of imminent failure.