Could harm plants but most fish are fine with excess tanins
If anything, the tannins help plants. The tannins, being dissolved organic compounds, tend to bind up with some of the minerals and heavy metals in the water. Specifically, they help bind up with micronutrients that the plants need, and in that bound-up state, plants can uptake those necessary micronutrients easier. It is all part of the cycle of plant life. When a plant dies and releases these tannins, it help the still living plants take up the necessary micronutrients. If you want to read more about this, please see Diana Walstad's excellent book
Ecology of the Planted Aquarium.
If you were thinking that the dark color absorbs too much light, that usually isn't of a great concern either because the specific wavelengths of light that most aquatic plants prefer aren't strongly absorbed by tannins.
In addition, because the tannins bind up with heavy metals, it also protects the fish from those possible dangerous heavy metals, too. Some fish will be more likely to spawn in tannin infused water, too.
In short, I think that tannins provide quite a lot of benefits and really no negative side effects at all, except some people don't like the look.