What are the networking requirements for a successful AirPlay connection?
Per Apple KB article HT6175, AirPlay makes use of a number of ports:


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║ Port ║ TCP/UDP ║                  Name                  ║
╠══════╬═════════╬════════════════════════════════════════╣
║   80 ║ TCP     ║ Hypertext Transfer Protocol (HTTP)     ║
║  443 ║ TCP     ║ Secure Sockets Layer (SSL, or "HTTPS") ║
║  554 ║ TCP/UDP ║ Real Time Streaming Protocol (RTSP)    ║
║ 3689 ║ TCP     ║ Digital Audio Access Protocol (DAAP)   ║
║ 5353 ║ TCP     ║ Multicast DNS (MDNS)                   ║
╚══════╩═════════╩════════════════════════════════════════╝