Iodine is an essential mineral in the diet due to its importance towards cognition and fetal development secondary to being required for thyroid hormones; iodine is central to the active thyroid hormones T3 and T4, and a true iodine deficiency results in less of these hormones and may result in reduced cognition (if a subclinical deficiency) or cretinism (severe deficiency in utero).
Despite the importance of iodine, it is not a common dietary supplement. This is due to table salt being iodized (added iodine) and even relative deficiencies being quite rare in first world countries (it is a common issue in developing countries due to iodine only naturally occurring from fish and seaweed which may not be consumed); actually benefitting from supplementation of iodine requires a 'perfect storm' of situations to occur which are outlined in the dosing section but not many people will meet these requirements.
Supplementation of high doses of iodine in otherwise healthy people does not appear to result in much, since it is readily excreted and normalized. There may be a very small and (clinically) irrelevant antiinflammatory effect and a small reduction in thyroid hormones (rather than an increase), but that seems to be it. Obscenely high doses for a prolonged period of time, which occurs with consumption of unprocessed seaweed (mostly kombu) will result in benign goiter in all persons and thyrotoxicity in some persons with underlying thyroid issues.