Proxmox has made this free utility to backup running OpenVZ containers.  It's a great program which is actually just a PERL script but gets the job done.  This program is not 100% required because all it really does is cp -a from your container's path as far as I know but it is still good to have uniformity to how you backup your containers.

For RPM distros such as Centos/RHEL/Fedora etc.. download and install this:

wget http://www.proxmox.com/cms_proxmox/cms/upload/vzdump/vzdump-1.1-2.noarch.rpm; rpm -ivh vzdump-1.1-2.noarch.rpm

For Debian based distros like Ubuntu etc.. download and install this:

wget http://www.proxmox.com/cms_proxmox/cms/upload/vzdump/vzdump_1.1-2_all.deb; dpkg -i vzdump_1.1-2_all.deb

vzdump options

 vzdump --help
Unknown option: help
usage: /usr/sbin/vzdump OPTIONS [--all | VPSID]

        --exclude VPSID         exclude VPSID (assumes --all)
        --exclude-path REGEX    exclude certain files/directories
        --stdexcludes           exclude temorary files and logs

        --compress              compress dump file (gzip)
        --dumpdir DIR           store resulting files in DIR
        --tmpdir DIR            store temporary files in DIR

        --mailto EMAIL          send notification mail to EMAIL.
        --quiet                 be quiet.
        --stop                  stop/start VPS if running
        --suspend               suspend/resume VPS when running
        --snapshot              use LVM snapshot when running
        --size MB               LVM snapshot size

        --node CID              only run on pve cluster node CID
        --lockwait MINUTES      maximal time to wait for the global lock
        --stopwait MINUTES      maximal time to wait until a VM is stopped
        --bwlimit KBPS          limit I/O bandwidth; KBytes per second

        --restore FILENAME      restore FILENAME

Real Life Usage To Dump/Backup an OpenVZ Container

*note that since I chose --compress (why not compress right?) that the file will actually be:

vzdump-5000.tgz

vzdump --compress 5000
INFO: Starting new backup job - vzdump --compress 5000
INFO: Starting Backup of VM 2000 (openvz)
INFO: status = CTID 5000 exist mounted running
WARN: online backup without stop/suspend/snapshot
WARN: this can lead to inconsistent data
INFO: creating archive '/vz/dump/vzdump-5000.dat' (/vz/private/5000)

To Restore The OpenVZ Backup

vzrestore /vz/dump/vzdump-5000.tgz 5000

*note that if you specify something different eg. 10000 as the VPSID it will restore to that container.