According to the DivaCup website, the menstrual cup should not interfere with internal birth control devices.
However, with an intrauterine device (IUD), this may depend on how long a person has used one. The United Kingdom's National Health Service (NHS) states that a person may accidentally pull the IUD strings out if caught between the vagina and the cup's edge. It recommends that individuals only use a menstrual cup 3 months after IUD insertion.