Bug 30950 – iphone: sync with shared contacts and calendars
The problem as described in the bug is that ActiveSync, itself, doesn't support shared calendars. It doesn't matter if you have an iPhone or Windows Mobile--if you sync using ActiveSync, you won't see shared stuff. Exchange or not also should not matter.
A separate issue is that iPhone is the only device that can sync multiple calendars over ActiveSync. Or at least it has been, maybe that's changed recently.
According to some comments in the bug, Google has some sort of workaround that allows viewing shared calendars over ActiveSync. Not clear how they do it, or exactly what the user experience is. See also
Choosing Which Calendars to Sync : Mail, Calendar, & Contacts - Google Mobile Help
For Zimbra as it is currently, the best workaround is to use CalDAV instead of alongside of ActiveSync. Depending on whether you need RW or only Read access, you might be able to use WebCal ("Add Subscribed Calendar" option under "Other" under "Add Account...").
Still another possibility would be to subscribe to the shared calendar within ZWC as if it were an external calendar ("Synchronize appointments from remote calendar" after creating a new calendar, using the shared calendar URL). This might sync down (read-only) via ActiveSync to an iPhone, but it can cause some wonkiness, like events not being rendered if they exist on both the account owner's calendar and the shared calendar.