This question doesn't duplicate the more limited one on Zicam; I ask here about zinc in general.
No: Mayo Clinic (from which I quote below) and WebMD are skeptical.
Since then, research has turned up mixed results about zinc and colds
Possibly yes: See NHS (from which I quote below) and The Telegraph .
For many years, it has been suggested that zinc supplements could relieve the symptoms of the common cold, but there was little clarity until a 2011 evidence review. This recent review suggested a modest reduction in the length of symptoms, although the results were not entirely conclusive. In a bid to help clarify the matter, researchers have conducted another review of evidence on this issue.