I've asked Ria Peters of Seaweed Ireland, and here's what she says:
We all do need iodine but some people are sensitive for excess and there are a few iodine-induced diseases as well. However, it involves a small group of people and iodine shortage is most common. If you are sensitive to an excess amount of iodine, the effect can be severe and dangerous.
For most people, it isn't a problem because the body normally can readily adapt to high iodine concentration.
Brown seaweeds in general are very rich in iodine with kelp at the top, red type seaweeds contain about 50% compared with the brown type, and greens are generally low.
Still seaweed can best be used as a supplement in small quantities because of the presence of lots of nutrients and minerals in concentrated form. If consumed in large amounts on a regular basis it can and will lead to illness.
It is still best to consult your health practitioner or your doctor for advise.
Learn more about seaweed and the thyroid from Dr. Ryan Drum and Dr. Stephen Langer.