Airplay does not work on Verizon FiOS

I have three Apple TVs. All of them work fine. However, I cannot connect my iPhone using airplay. I constantly have to reset the Apple TV. It will work for a few days and then stop working. When I open AirPlay from the control center the TV is there. When I tap on it to connect, the check mark appears indicating that it is connected. But, within a second it disconnects. I have tried renaming the TVs, editing the security settings, disabling the IGMP protocol and all the obvious fixes. None have been effective. I'm using Verizon FiOS with their latest router. I'm at my wits end! Does anyone know why this happens and/or how to fix it?