Since the "upgrade" to Sierra, AirPlay no longer works properly with my Apple TVs. I have the same problem. The Macbook is OK (mid 2011). The Apple TV is OK 3rd generation updated. The Time Capsule (network) is updated too.
The devices are on the same network (checked many times!). The Airplay monitor cannot be activated (the only option is 'not activated'). If I click in the bar on the 'monitors icon', the Mac do not find the Apple TV and the message appear 'deactivated'. But on the television (B&O BVI 11-46T2) the menu indicate 'Apple TV activated'. Everything seems to be OK. OS Sierra may be not?

I have a similar problem trying to connect my 2 iPhones, my iPad and my iPad mini. May be there is a problem with iOS too? With these devices I can play music from music.app or a video from VLC.app : sound is OK but no image appears.
iPhone4s, iPad and iPad mini have iOS 9.3.5 and they can detect Apple TV:


But iPhone5s with iOS 10.1.1 cannot detect Apple TV. The device remain on searching for Apple TV...:

On the television, the Apple TV is 'activated' as this appear on the screen:

After hours of work on this subject, it remain the same. Very disappointing!
If somebody have a solution, many thanks in advance!