Can network settings or a firewall contribute to or resolve Diablo 4 Error 316?

Yes, network settings and firewalls can definitely play a role. A restrictive firewall (Windows Firewall or third-party) might block the Battle.net client's communication with Blizzard's license servers. Ensure that Battle.net and Diablo 4 are allowed through your firewall. You can also try flushing your DNS cache (ipconfig /flushdns in Command Prompt) and renewing your IP address (ipconfig /release and ipconfig /renew) to resolve any local DNS resolution issues that might prevent license verification.