The phrase "closing door on" can be substituted with a variety of other terms, depending on the context. For instance, you could use "shutting out," "excluding," or "closing off." Other possible alternatives include "blocking access to," "preventing entry," and "sealing off." These phrases can all be used to describe a situation in which someone is denied access to something or is prevented from participating in a particular activity. Whether you're talking about a physical barrier or a metaphorical one, there are plenty of synonyms for "closing door on" that can help you convey your meaning more effectively.