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.
Supplementation of iodine is designed to circumvent a deficiency, and deficiencies of iodine are quite rare in first world countries. For those in a first world country, iodine should only be considered if you meet all of the following requirements:
You are a vegetarian or vegan who actively avoids processed foods, or a meat eater who never eats fish and avoids processed foods
You avoid adding additional salt to your diet
You avoid consumption of seaweed or seaweed based products (such as sushi, which are wrapped with Nori)
(Any of you bodybuilders meet those criteria? I did.)
The recommended upper intake is in the range of 500-1,100μg when looking at international bodies, while the highest recommended upper limit is 3mg; consumption of seaweed routinely is thought to put people near the highest upper intake level