Fill the pitcher with hot water and drop in several denture cleaning tablets (use the cheapest kind you can find). Let sit overnight. Next day, pour the water out and wash the pitcher with a rough wash cloth (I save old ones for things like this). If the stains are built up you may have to do this several times. After that do it once a week for maintenance.
I used to work as a waitress and we cleaned the glass containers with ice and salt placed directly in the container. You shake the container and roll it gently in your hands so the ice moves around with the salt. It worked well. All you do is rinse when complete.
I make several pots of tea every day. The easiest way to clean the pots and pitchers is to sprinkle baking soda and salt inside then use a sponge [reserved just for this use] to wipe off all the stains. Use the hottest water from your faucet to thoroughly rinse. Everything is naturaland inexpensive, too.
Also make a paste of baking soda and a few drops of water. Rub with cloth, tea stains will come right off.