I also supplement with iodine, and was loothe to go off it during the MC, but Burroughs says to avoid all supplementation and medicine (unless prescribed by a doctor) so as not to interfere with the MC.
Another point is that Iodine
directly affects the thyroid gland, at the base of your neck. This gland controls metabolism. Your body is already fairly stressed by the detoxing/cleaning/rebuilding efforts of the body. It's best to leave all optional supplementation, particularly that which might affect digestion and metabolism, for after the fast (and I'd suggest after the minimum 4-day recovery period).