Are there any side effects to taking zinc supplements?

Take in moderation. Too much zinc can prevent absorption of other elements such as iron and magnesium.
Nausea. Far and away the most common side effect of zinc supplements is nausea. Like all minerals, zinc is better absorbed on an empty stomach but some people will get nausea from taking much on an empty stomach. If this is the case then take it with food. You could take a higher dose if you take with food to compensate for the lower rate of absorption.