Do you mean upgrade Zimbra or upgrade clamav? I meant to upgrade your clamav from the source package. I had a very much similar problem to you, I had just installed 4.5.6 (in fact I was having the same problems with 4.5.5) and the antivirus wasn't starting. I had tried various things and occassionally when I tried to restart zmclamdctl it would try to access the site to update the virus definition and have problems.. I'm not entirely sure if our problems are the same but I thought maybe updating the clamav version might be a good thing to try for you.
My Zimbra version 4.5.6 came with clamav-0.90.2 you can see it in the /opt/zimbra directory. The latest version is clamav-0.91.1 ; I followed the "updating clamav" wiki link
Updating CLAMAV - ZimbraWiki . There were a few extra things I had to do with the Ubuntu 6.10 server that weren't in the guide (it is a general guide);
apt-get install zlib1g
apt-get install zlib1g-dev
apt-get install build-essential
apt-get install build-dep clamav
However I might be wrong and you meant you updated the Clamav version?
It seems to me it's a bit of drastic measure to go to.. but it was quite interesting to do and get it working finally.. and Zimbra seems such great software. All the services seem to be running fine at the moment. I will let you know though if it breaks again.