It's got to be the easiest way to clean a teapot stained with black tea. Here's my teapot, which serves us well making black tea for us every afternoon.
Make a pot of fruit tea. I used some old loose fruit tea that is so old I don't feel like drinking it anymore. But it would work just as well to make a pot you want to enjoy too. After the tea has sat in the pot for a while, empty it. Then when you clean it, the stains will come off with almost no scrubbing. I used warm water, soap and a sponge, and just rubbing gently gave that distinct line you see in the right photo.
Rinse it out, and it's good as new!