Tea bags can be used one or two times. After that the tea may lose some of its flavours and strength.
You'll need to store the tea bag in a moist medium. Once they get dried out, they become a breeding ground for mold and bacteria. So your best bet is to store them in a small shot glass filled with water. That will keep them seeping while they are being stored. While you can leave them out in your kitchen, I usually put them in the fridge so that there is less chance of bacteria growth. If you think you've got bacteria or mold in your tea bag, the best indicator is smell. If it smells funky, trash it.