If your Ubuntu VPS has seemingly lost all network connectivity after a reboot, login to the VPS via VNC and run the following command:
mv /etc/network/if-pre-up.d/iptables /etc/network/if-up.d
The cause of the problem is that the network startup procedure will load the script /etc/network/if-pre-up.d/iptables before the network service was loaded, and then fail, causing networking to be off on the VPS upon boot.
Thank you to Zihao Lee for contributing this solution.