I have an OpenVZ configured with HA, live migration, backup using suspend though but suspension is only a matter of seconds and provided we are talking about a webserver, ftp server or the like a few seconds suspension does not matter much. The same applies to online migration. The two most important part in downtime under backup and migration is read/write sped on storage and bandwidth on network. Provided at least gigabit network and a descent SAN or NAS you should be able to use OpenVZ for HA when the service can tolerate a few seconds downtime under backup and live migration.
For the backup or live migration to work no console must be active or an error like this is shown:
INFO: Can not suspend container: Resource temporarily unavailable
INFO: Error: foreign process 10221/971548(vzctl) inside CT (e.g. vzctl enter or vzctl exec).
INFO: Error: suspend is impossible now.
Requirements: Your storage must be on a shared NFS or SAN.