Note: Claims in this article and their associated citations can be found on the page of iodine, in the first major section titled "Sources and Status"
Seaweed products are frequently recommended for their health promoting properties, due to having a large Iodine content and some other beneficial components only found in seaweed including Fucoxanthin, fucoidans, and phlorotannins.
That being said, when we mean a large iodine content we mean very large. Insofar as that even the lowest source of iodine from seaweed (Nori) at the lowest detectable estimate (12µg per gram) is enough to reach your daily requirements when consuming 9 grams of the seaweed product only.
The largest source, Kombu (highest estimate being 2,660µg per gram) consumed at the same 9 gram dose would result in you reaching the daily recommended intake about 240 times over and exceeding the highest known tolerable upper limit by 800%.
While not acutely lethal and the actual 'toxicity' being dependent on underlying thyroid disorders, such high levels are known to acutely suppress thyroid function and eventually cause goitre.
At even small levels of dietary intake, depending on the source of seaweed you could be exposing yourself to toxic levels of iodine intake. It is not lethal nor acutely toxic, but could be toxic over the long term in some persons with underlying thyroid disorders and in all persons can cause goitre
There are a few dietary seaweeds that are most common:
Kombu, also known as Kelp; this refers to the genera of Laminaria
Wakame, which refers to the genera of Undaria
Nori, which refers to the genera of Porphyra
In the above scenario, Nori is almost never a rich enough source of iodine to pose a significant health risk. Wakame is somewhere in the middle (where moderate consumption should be fine, but excessive consumption of over 10-20g daily could cause issues) and Kombu which is a significant risk for iodine toxicity.
Keep in mind that, on average, the Japanese diet contains around 5g of seaweed daily and iodine in the range of 1,000-3,000µg.
Moderate intakes of Kombu may post a concern, whereas high intakes of Wakame may also pose a concern. Nori, the seaweed used to wrap sushi, does not pose a significant concern due to relatively low iodine concentrations
Boiling seaweed in water for 15-30 minutes appears to be sufficient in reducing iodine content in the seaweed (as it is leeched into the water and iodine may be released into the atmosphere due to it being a gas in its natural state), and this processing may eliminate up to 99% of iodine from Kombu (other seaweeds may not have such a high loss).
Due to such high losses of iodine from seaweed, usually processed or cooked seaweed does not carry as significant a risk of iodine toxicity (such high risk being seem with raw kelp products).
Boiling or heat treatment of seaweed can eliminate a great deal of iodine from seaweed, insofar that the toxicity issue may only exist with raw kelp
Goitrogens are compounds that are known to have anti-thyroid properties, and in some developing countries goitrogens (usually from cassava) augment iodine deficiency by competing with what little iodine is being consumed.
The competition between goitrogens and iodine is bidirectional, so in instances of excess iodine it is thought that goitrogens actually prevent toxicity from occurring due to preventing their uptake a bit.
Interestingly, many traditional Japanese dishes (such as soups) that contain seaweed also contain foods with a known goitrogen content such as soybeans (source of Soy Isoflavones), broccoli, and bok choy.
Coingestion of seaweed products with goitrogen containing foods may provide a possible protection against high iodine intakes