If you keep small birds as pets, you may have heard that you shouldn’t use galvanized hardware cloth when making your own birdcage. The reason is that high levels of zinc can be toxic (this is true for humans too). When hardware cloth is galvanized, sometimes little globs of it will be left near where two wires cross. Small birds that use their beaks to climb and maneuver around their cage can eventually break these off and swallow them whole.
This is not a problem with chickens, however, both because of their much larger size and because they don’t climb or chew on the cloth with their beaks.
What about plants? Plants actually need zinc, though as with anything, too much can be harmful. You’ll notice with your hardware cloth that, over time, the finish begins to get duller. This is the zinc gradually weathering off the wire, washing down to the ground. This process happens so slowly that it should not create a harmful buildup of zinc in the soil.