Don't let means don't allow in this case. Don't leave me means don't go away. Don't leave it means don't leave it here, take it with you. (this could have other meanings but I think this is the sense you want).

Don't let me. (It means you don't allow or permit me to do that) Don't let it. (It means someone asked you not to let it...for example: not to let that happen) Don't leave me. (It means I asked you to stay with me or not to go away from me). Don't leave it. (It means I asked you to take it away with you)