All of our used tea, both the bagged kind and the loose type, gets tossed into the compost bin.
I realize this is a plant-related site, and not a furniture renovation site,
but one thing you can do with a used tea bag (regular type tea) is to place it on a piece of scratched furniture to make the scratch look less noticeable. Mom used to do this to the wooden banister because we were forever scratching the wood. Afterwards, my dad would bury the tea bags along with the other compostable items in a shallow hole in the garden.