Hi Dave!
The thyroid is the master gland for your body's metabolism. Low thyroid function means lower metabolism, including the digestive system and production of stomach acid. Low stomach acid is one of the causes of reflux. The stomach has to churn more in order to get the same amount of work done. This is also why a lot of hypoT folks are chronically constipated - the digestive system slows down.
I also take iodine supplements. I've been on 8 drops of 5% Lugol's solution (the liquid equivalent of Iodoral) for a couple of months now. While I noticed some improvement with reflux since taking it, the biggest improvement has been from removing all gluten and dairy from my diet. While other people have also noticed improvement with their reflux while taking iodine, its usually not the primary reason they take it.
I've read (I think) that you're taking Iodoral? How much are you up to a day? Have you been tested for hypoT? What kind of symptoms are you having? Just so you know, if you have thyroid antibodies, iodine may not be the best choice for you as some folks with Hashi's get a bad reaction from it. Also, did you know there's a complete protocol to follow when taking iodine supplements?
Hrrrm. I think I just asked more questions than I answered. Anyway, if you want more information, feel free to ask.