Some cup companies suggest against using a menstrual cup with an IUD. However, many people are able to use both at the same time. I would contact the company if you’re are unsure.
I don’t have an IUD so I can’t comment through experience, but I have heard that if notify your physician that you plan on using a menstrual cup, they can cut the “strings” shorter so that they don’t interfere or get tugged on while removing.
Here is a list of cup companies which I have found to include information about IUD’s and Contraceptive Rings.