Does the tannin level in plants go down or up after they die? Or does the tannin level stay the same regardless?
Well maybe I'm confused, but do plants not produce tannins? If they do
  does the production of the tannins stop after death?


That depends on how you define death. Tannin is secondary product, so I don't think the plant cells will produce more tannin when they are starving probably because of cutting off the leaves, roots, etc... when the plant dies. The plant won't bother to degrade tannins (I guess it does not even have the pathway necessary to do that), because it is a defense mechanism against insects, mammalians, etc... and not an energy storage material. Tannins can be degraded by microbes, heat, UV radiation, chemical reactions, etc... So if you can't protect the tannins from all of these then they will degrade. (Note: you can't, all you can do is slowing down the process.)