Resolution
The mailing may not be set up to send tracking links or the open tracking image, or there may be an issue linking back to ListManager from the mail message.
First, confirm that the links are tracking by making sure the links are pointing to http://listmanagerserver/t/#/#/#/#/ instead of directly to the final location. After confirming that the link is pointed back to ListManager, verify that the link loads from email messages sent to third-party locations. If the links load up through the tracked link, then ListManager will log the clicks.
If the links do not point back to the ListManager server, verify that the URL to web interface setting inside of Utilities: Administration: Sites is pointing back to the ListManager server. If the links work on an internal network but not the external network, check to make sure the firewall allows the connections through.