If it`s a metal teapot add some washing powder to the pot of water and bring to the boil. You may have to do this a few times but it does work. If it`s a china teapot an equal mixture of salt and white vinegar will remove the stains according to the book 1001 practical uses for vinegar. Failing that, get a new teapot! I`m disposing of mine as I never use it anymore. One careful owner and spotlessly clean, the teapot and the owner!
I have used" Vanish" in my china tea pot and it came out spotless.
I put about a tablespoon of Vanish into it and added boiling water so it came to the top of the spout and left it overnight.