I was diagnosed with Hypo in 2006 and was on levoxthyorine for 10 months and took a supplement with Iodine. In Jan 2007 I was diagnosed with Hyper/Graves. My doctor didn't think the iodine was a factor and I do have a history of graves in my family- but I also wonder if the excess iodine impacted my thyroid function. Tricky little gland...
I would still have regular tests until the levels straighten out for a long period of time--over two years. After that, if normal, I would still be checked annually.
Yes, too much iodine in the diet is not good for the thyroid. You get more than enough iodine in the typical American diet. If you are really worried about a lack of iodine, then prepare some meals with iodized salt and some without. Actually, the body does not need a ton of iodine to be healthy. Fast food is full of the stuff and is way over the limit most people need in a typical day--or week for that matter.
I can tell you that when I started out being hypo, I did take iodine and tyrosine and became very hyperthyroid but I was also over medicated. I believe if I had not taken the iodine and tyrosine, I would not have gone hyper as fast as I did. I am back to hypo now and do not take iodine or tyrosine anymore. I feel if I did, I would go back to the same predicament I got into before. Just my experience.