This information is contained in the README.txt that is included with the installation download, but I just spent a long time looking online for any information about updating (or upgrading) ZCS Open Source Version and found a severe lack of instructions for reference. Even though this is exceedingly simple I thought I would put it up here.
If you are planning to update your OS (say, from Ubuntu 10.04 to Ubuntu 12.04) then this is not the process you are looking for. This is the process for updating/upgrading ZCS without updating/upgrading your distro at the same time. That process is all over the interwebz and can easily be found.
Download the version you want to update to for your distro from here.
While that is downloading, make a backup of your existing Zimbra directory.
Now that you’ve made a backup (you made a backup right?), unpack the new ZCS you downloaded.
cd into the unpacked directory and run ./install.sh
Answer the questions you are prompted to answer, and if all is well, you will be asked if you want to upgrade. If you answer yes, your Zimbra services will be shut down and upgraded. You will not lose any data from this process (but you have a backup just in case, right?).
That should be all there is to it. Enjoy your new features and security!
edit: fixed the link