Can I mean "May I leave the door open?" by asking "May I let the door open?" in any context? I know that there is, of course, a difference in meaning in both of the sentences, but I'm just wondering if I can ask "Should I let the door open?" to mean "Should I let the door at its default(open) position?"
If the door is currently open, whether or not it would return to the open position by default, you would say, "May I leave the door open?"
If the door is closed or nearly so, then you can say, "May I let the door open?" Basically, substitute "allow" or "permit" in the sentence; "May I allow the door to open?" only makes sense if the door is not already open.