Many types of black and green tea have natural fluoride in them. Fluoride will block iodine absorption in the thyroid. Perhaps if you take a daily dose of iodine it will be enough to prevent this isssue... relying on just what is in food or salt will surely keep the much of this tiny amount of iodine out of your thyroid.
Because Japanese have historically have had such good thyroid health and they drink lots of black and green tea, but also, if on a traditional diet, eat 12.5 mg of iodine a day, I would say that the two can work together if there is enough iodine in the diet.
"Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth." Marcus Aurelius