In general I would take loads of vit C if my gums were inflamed. 5-6 grams a day, divided.
Iodine doesnt cause inflammation, but it may put a demand on the body; for certain nutrients / vitamins / minerals.
Adrenals may be involved in this; and probably need vit c and b more than ever to keep up with the awaking thyroid. http://www.drlam.com/articles/oral_health_and_vitamin_C.asp
Make sure you take all the co-nutrients like magnesium, selenium, vit B and C, salt etc; that follows the Iodine protocol. Dont cut short on these.
You are early into the Iodine adventure; it will take some time, your body will give various signals and symptoms, depending on your health challenges.
A rough phase up to 2-3 months may be expected, not all go through this, but some of us do.