Fantastico F3 doesn't require many ports to be open, but here is a complete list, so you can make sure your firewall is configured appropriately

Port 80: Open inbound and outbound (TCP/UDP).

The following ports only require LOCAL on-server access for API use on servers with cPanel:

2082, 2083

2086, 2087

That's all. If you get an error while trying to use Fantastico F3, please make sure the ports listed above are open as appropriate).