Your teapot is not like any other teapot. The materials and shapes can vary immensely. Porous teapots for example, like those made of clay, should be rinsed out with hot water only, without the use of detergents, as these settle in the unsealed surface of the teapot and could thus affect the tea’s flavour. This goes for the tea’s innate flavour as well, as it penetrates the pores and enhances the flavour: a sometimes desirable side effect. Therefore, you should only use one tea per teapot.
Glazed teapots are a different story. Thanks to the finish, the sides of the teapot do not absorb unwelcome flavours. With time, a dark patina forms on the sides of the teapot. If you use black tea much, the patina will form very quickly, as this kind of tea contains many colouring substances. It is not necessary to remove or clean the patina. It is, however, important to leave the teapot to dry without the lid to avoid germ formation. If you do not like the dark patina, there is no need to resort to aggressive detergents.
The dark coating can easily be removed using lemon or baking soda. Fill your teapot with warm water and add either the juice of half a lemon, or two spoons of baking soda. Stir the liquid to mix the components and let it work in overnight. Rinse the teapot well in the morning and the dark stains will disappear.
In general, you should not use rough cleaning materials. When cleaning iron teapots especially, a rough sponge could damage the fine enamel coating which protects the teapot from rust.