That highly depends on the underlying storage. If it is something that supports snapshots (ZFS, Ceph, LVM thin) then it should work fine, also backups will be possible without any downtime as they will be read from a temporary snapshot.
But even if they had, the RAM snapshot needs to be written, but without freezing the container. I would appreciate an option when I could ignore everything that was not fsyned, e.g. Postgres use case. In that case the normal ZFS snapshot should be enough.
The state of a container is not part of the snapshot (just checked), as it is really hard to capture the state of the container (CPU, network, all kinds of file and socket handles) and restore it because all an LXC container is, is local processes in their separate cgroups. This is also the reason why a live migration is not really possible right now, as all that would need to be cut out from the current host machine and restore in the target machine.
This is much easier for VMs as Qemu offers a nice abstraction layer.