Sounds like you either don't have read permission to c:\FullInventory.zip or write permission to C:\. This has nothing to do with DotNetZip and is an issue with the Windows user account and it's associated file system permissions.
I also find it easier to use this instead of a stream, unless you need the stream:
var filePath = @"c:\FullInventory.zip";
using (var zip = new ZipFile(filePath))