As long as you’re careful to avoid suction or pulling your IUD strings when you remove your menstrual cup, it’s fine to use a cup and an IUD together. A recent study found that the “use of tampons or menstrual cups does not increase the risk of early expulsion of an IUD.” The packaging information for IUDs can be inconsistent, but women “can use whatever their usual menstrual product is and not increase their risk of expelling the IUD.”
Menstrual cup brands are in agreement about this. DivaCup, MoonCup/Keeper, MeLuna, Lunette, and SoftCup say that because of the physical placement of your cup, you should be fine combining one with an IUD. If you can, try to double check with your health care provider first to be on the extra-safe side.