I'm not ready to totally dismiss it, only had asked for some evidence other than anecdotal and none was forthcoming.
Dr. Emoto taught we can alter the structure of water with thought. Since these iodines are aqueous, it might just be the people putting the magnets on the bottle, are thinking "this will make the Iodine better" , and those thoughts structure the water, not the magnets.
I don't know. One possibility is to do a double-blind taste test, using magnets, no magnets, and also strap some other object to the bottle, like a non-magnet but tell the person its a magnet.
But who would care ? lol. I have better things to do, than even to be writing this message. Be well