Whatever you do, don't throw them away once they're used. If they're the kind with strings, pull the strings off and use them to tie things together, hang decorations in a window (with a thumbtack) or put a couple dozen together, tie in the middle, trim and you have a cotton pompom. Tea bag strings can be dyed easily with generic kool aide, tea or coffee. Don't make up a packet of kool aide for that purpose, but when you're making it anyway, take out a tablespoon or two before you add sugar and dye the cotton string with that.
Oh, and if you soak the tea bag strings in borax and salt dissolved in water then let them dry for a few days, they make good candle wicks for short candles.
I'm not through yet. Use the tea and the bags themselves to compost, or dry them completely and throw them in the fireplace or wood stove. Put them on your houseplants before you water them.
Just because tea bags are used, doesn't mean they're not still useful. Now, go apply that to everything you can think of.