No. LinkFixer Advanced does not required a connection to the Internet in order to be licensed successfully or to process files and links. It is, however, recommended.
LinkFixer Advanced can be licensed and run in "off-line mode". For more details on the "off-line" activation process, see the knowledge base article located here: https://linktek.freshdesk.com/solution/articles/31000014830-my-computer-is-behind-a-firewall-with-no-internet-access-how-do-i-activate-the-software-.