This came up a while back and from what I could find the animal tests didn't indicate significant effects until quite high dosages taken continuously, then again I believe hormones and other important blood constituants can be depleted with long term use, which may not be significant in the short term, but of course it depends on the individual,
If you take a Iodine supplement that could offset any slight thyroid hormone effects and the body adapts to compensate also but I still wouldn't take Miracle-Mineral-Supplement longterm unless I could find a very good reason and no alternative.
I doubt chlorine dioxide has any displacement effect on Iodine and the way it affects Iodine in the GI tract and thyroid hormones isn't certain either