Most major brands of tea use food-grade plastic for the bags. Such tea bags are not biodegradable. If you prefer tea bags made of paper,  check online for a list of plastic-free teas available locally

Tea is, of course, biodegradable. But plastic is not. So if you have a composting system and you throw the entire tea bag in there, in a few months you will find a perfectly intact, but empty, tea bag. Fun! 

You can achieve the same effect in a garden pot with soil but it might take a bit longer depending how active your soil is.