Just tried something and now I'm very confused.
You mentioned permissions so I decided to log in as the 'admin' user and do some testing there. The drag functionality works as expected.
I then created another user account and configured for email. Logged in as that user and attempted to drag a freshly created calendar entry with no luck.
Here is where it gets confusing (for me):
I create a new account with no email setup. Create a fresh calendar event and dragging works perfectly.
Perhaps some background information on my email setup will help:
When this server was initially set up, I had attempted to set up some external accounts. There is a current bug in Zimbra 6.0.3 which results in the following error whether you set the account to leave messages on the server or not:
Error: leaving messages on server requires support for UIDL
Here is a thread discussing the issue:
Error: leaving messages on server requires support for UIDL
Due to this problem, I implemented a workaround or sorts to get this working for me. I created a Persona for my ISP address and enabled the option to send email from any address. I then set all reply-to preferences to show my ISP email address and set the Persona as the default address to send messages from. Email works great. Sending and receiving have been flawless. The only issue I can see is the calendar drag function being killed in those accounts that have the email set up.
To further test this, I've removed the Persona on my account. As it turns out, it looks like that wasn't needed in the first place. Simply having the ISP email address in the "Choose the name that appears in the "From" field of email messages" section provides the functionality I need.

Dragging calendar items still doesn't work though.
I'm playing with the mail config right now, flipping options on and off to see what the trigger is. I'll report back if I find the cause.