The DivaCup website assures women that it's totally safe to use the DivaCup with the IUD, but that's just not true. At least not for everybody, and they should issue warnings, just like for all other risks.
Have you had any problems with the DivaCup while using an IUD? Have you told DivaCup?
EDIT: Good news -- the IUD just moved, but not enough to have to be removed.
The bad news is that this ends any more cup use for me. Even though some women can use both, I'm not willing to risk it. (though I might try modding the IUD to make it shorter/less suctiony -- I'll let you know if it works, or just destroys it completely)