Q: Ports required

If it's not on the same network, it won't work. What ports are open on the firewall are irrelevant.

Is there a router between the Apple TV and the machine containing iTunes you're trying to Airplay from?

Apple state that it needs to be on the same network in the requirements & troubleshooting that I have seen.

If you have it working on another network you will need to review what is setup there. FYI Airplay relies on bonjour to discover the devices. That does not cross between networks unless you have a router(s) that handles it for you.

I suspect you need to explain more about your networking setup. Port forwarding may not be what you need to do.

There are pages that cover the topic if you look…

http://irq5.io/2011/01/02/mdns-repeater-mdns-across-subnets/

Bonjour does NOT work across networks. It can not be routed. It is strictly a local protocol.

See if bonjour browser helps…

http://tildesoft.com

The services will be advertised with port numbers.

This is an unofficial guide to Airplay too…

https://nto.github.io/AirPlay.html

I don't think Apple have documents online for it.

I am VERY familiar with enterprise WiFi networks. I am not familiar with your network. Perhaps you should talk to your network engineers about this problem.

TCP and UDP ports used by Apple software products - Apple Support

That page lists port 80, 443, 554, 3689, 5353 alongside 'Airplay'.

Some others are listed in the "Unofficial AirPlay Protocol Specification" I linked earlier. Search that page for 'port:' to jump to them. You may have to fire up tcpdump or maybe http://www.tastycocoabytes.com/cpa/index.php

Good luck.