Zinc is used to galvanise metal to prevent rust. Both, when ingested in sufficient amounts by a bird, will cause toxicity. Affected birds will be depressed, fluffed up and often incredibly thirsty, but vomit soon after drinking. Death can occur as little as three days after being exposed to a supply of lead or zinc. This is usually due to kidney failure, but other organs, including the brain, pancreas and intestinal tract can be affected.
Prevention is obviously the better way to go.
Firstly, wherever possible avoid cheap wire, as it often has tags and flakes of zinc on it, and contains high levels of lead. Use some of the newer wires by BHP that are welded after galvanisation. Yes, it does cost more, but the money spent on treating sick birds usually exceeds the difference between good quality and poor quality wire. If you can't get this wire, you must physically scrape off all tags of zinc, then wash the wire with dilute vinegar. Rinse off the vinegar after an hour, allow the wire to dry and then paint with a matte black paint.