I haven't noticed any staining yet. At 12.5 ppm it likely takes a long time to stain anything. And even if it does, it shouldn't affect your equipment's continued usage. The key is to prepare it at only the required 12.5 ppm. Anything more than that may not only stain, but compromise the no rinse nature of Iodophor. Per the podcast it is not at all to be considered "no rinse" at 25 ppm and above, so be very careful to monitor your concentration.
Since listening to the National Chemical CEO's podcast and realizing that I only need to prepare enough solution to wet the surfaces of my equipment, I only make up a single quart of 12.5 ppm Iodophor solution at a time now instead of my former standard practice of making up 5 gallons. I add 0.75 mL of Iodophor to 1 full quart of water. A very light straw color is the result.
I also emailed them and asked about the common practice (formerly mine also) of adding a capful to 5 gallons, and they said this results in a concentration somewhat above 12.5 ppm, but is likely still no rinse safe. They also commented that over time they have changed, or may in the future change their packaging and cap sizes, and that different volumes purchased come with different size caps, so they were not recommending this practice.
PS: I just tried the podcast again, and it still works fine for me.