Since iodine supplementation directly effects TSH, t4 and t3 levels (reverse t3 and t4 are also common thyroid tests), you may both see a high TSH and low TSH.
If I were you, I'd seek out an iodine-educated doctor when considering getting those tests done - because if it's a high TSH you're likely to get a prescription for thyroid hormone (yeah, they do sell their own version of iodine), and if you have a low TSH, they may consider you hyperthyroid and try to intervene with thyroid treatment to lower your thyroid hormone production.
What do you want to get out of the testing?
The longer you've been supplementing iodine, the more adapted your system will be to the supplementation.
If you're testing for progress on a hypothyroid status you've been working to correct through iodine supplementation -
remember that parasite cleanses are reccommended first. Lingering infections also effect thyroid hormone (t3 and t4) levels.