She doesn't like my being her superior. -- It's most correct to say "my" and not "me" but you will certainly hear "me." It means she doesn't like the fact that you are her superior.
She doesn't like me to be her superior. -- This is not a very likely sentence. It sounds like she has various people who are her superior; sometimes it's you and sometimes other people. She doesn't like it whe
How often is someone a person's supervisor sometimes, but not others? That's what would be required for "She doesn't like me to be her superior" to make sense to me. (She doesn't mind it when Joe or Mary are her superiors, but she doesn't like me to be her superior.)