Hello Gonzalo,
If someone says to you "don't dear me", it means they don't like how you are addressing them and would prefer less familiarity. As Joshua points out, it can be a bit hurtful. That's why it is a safer option to say "don't call me dear".
Similarly, someone could say to you with a friendly smile "don't Sir me" or "don't Madam me". In these two cases, they usually will add "Please just call me X (John, Sarah)". This means they want you to be less formal and more familiar.