The only solution after trying many suggestions was to simply restart the "vz" service (definitely not preferable).
Even doing an ifdown and ifup for venet0 did not help.
ifup venet0
Bringing up interface venet0:
Configuring interface venet0:
net.ipv4.conf.venet0.send_redirects = 0
RTNETLINK answers: Network is down
Some errors from log:
CPT ERR: d2dc6000,54819 :Failed ot lookup path '/dev/shm'
CPT ERR: d2dc6000,54819 :Failed to restore mount point: dev 'tmpfs', type 'tmpfs', path '/dev/shm'
CPT ERR: d2dc6000,54819 :rst_namespace: -2
=====================
CPT ERR: f3f70000,30222 :rst_file: -2 98752
CPT ERR: f3f70000,30222 :rst_files: -2
CPT ERR: f3f70000,30222 :make_baby: -2
CPT ERR: f3f70000,30222 :rst_clone_children
unregister net_device: 'tun0' not found
unregister net_device: 'tun1' not found
unregister net_device: 'tun2' not found
unregister net_device: 'tun3' not found
unregister net_device: 'tun4' not found
kobject_add failed for tun0 with -EEXIST, don't try to register things with the same name in the same directory.
[<c04eec02>] kobject_add+0x141/0x167
[<c0561c1c>] class_device_add+0x9d/0x3b4
[<fc1ff49e>] __ve_dev_move+0x17f/0x1b4 [vzmon]
[<fc1ff534>] fini_venet+0x61/0x81 [vzmon]
[<fc200731>] env_cleanup+0x3e/0x185 [vzmon]
[<fc2008a0>] vzmond_helper+0x28/0x34 [vzmon]
[<fc200878>] vzmond_helper+0x0/0x34 [vzmon]
[<c062d243>] kernel_thread_helper+0x7
CPT ERR: c02fa000,73 :CPT: lock fd is closed incorrectly: 1
Starting container...
Container is mounted
Adding IP address(es): 192.168.5.2
RTNETLINK answers: No such device
vps-net_add ERROR: Unable to add route ip route add 192.168.5.2 dev venet0
Container start failed (try to check kernel messages, e.g. "dmesg | tail")
Killing container ...
Container was stopped
Container is unmounted
Solution
service vz restart
openvz, cpt, err, fa, fd, incorrectly, vps, net_add, unable, ip, dev, venet, suggestions, restart, quot, vz, preferable, ifdown, ifup, interface, configuring, ipv, conf, send_redirects, rtnetlink, errors, dc, ot, lookup, shm, restore, mount, tmpfs, rst_namespace, rst_file, rst_files, make_baby, rst_clone_children, unregister, net_device, tun, kobject_add, eexist, register, directory, eec, class_device_add, fc, ff, __ve_dev_move, vzmon, fini_venet, env_cleanup, vzmond_helper, kernel_thread_helper, container, mounted, adding, es, kernel, dmesg, unmounted,