Iodine supplements usually aren't necessary if you live in the United States or in most developed countries. Some alternative medicine practitioners recommend iodine tablets or kelp supplements — which are high in iodine — for people with hypothyroidism.
It is true that iodine deficiency can cause hypothyroidism. But iodine deficiency is rare in the United States and other developed countries since the addition of iodine to salt (iodized salt) and other foods.
If iodine deficiency isn't the cause of hypothyroidism, then iodine supplements provide no benefit and should not be used.
In fact, for some people with abnormal thyroid glands, too much iodine can cause or worsen hypothyroidism.
Hypothyroidism can be safely and effectively treated with the synthetic thyroid hormone levothyroxine (Synthroid, Levothroid, others).