You sure can with Apple TV 4 with tvOS 10 or higher. Neon supports AirPlay from your iPhone, iPad or the browser extension on your Mac.
Make sure your Apple TV & device are on the same wireless network.
- Go to your TV and select the source your Apple TV is plugged into
- Select a show/movie on your device and press play
- Press AirPlay and select Apple TV.
Whatever you chose to play should now be on the big screen!