Medical questions posted on boards like UKP invariably yield anecdotal responses..
Don't bother with Zinc supplements - if you want Zinc, eat some oysters or similar.
The risk with supplements is that they are given in doses that are far above the safe upper limits you should be taking. For most water soluble vitamins, this isn't really a problem, as it washes out of you quickly - Too much niacin will make you itch for a few hours, too much C might give you a stomach ache, etc. But for the fat soluble vitamins and minerals (such an Zinc), you risk serious health complications, especially if you are taking high doses (what is typically contained in a supplement tablet) over many days. The shortest term risk is that taking high doses of Zinc - what is found in a typical pill - precludes your body from absorbing Iron. Iron is necessary for countless body tasks, nevermind making up the heme portion of hemoglobin, which your body uses to bind oxygen to circulate to your individual cells. Taking high doses of Zinc has other numerous potential problems which are beyond the scope of what anyone on this forum wants to read. The good news: The forms of Zn contained in most supplements are very poorly absorbed, so most likely if you take them, you're just wasting money - unfortunate, but not necessarily unhealthy. But check the label - you really don't want to exceed 20mg / day, with perhaps 40mg being the upper limit.
It's ALWAYS better to get your vitamins / minerals by eating a wide variety of different foods - in this way you'll assure you get everything and overdose on nothing.