This does not work because the Tivo App does not register with the new UI and I imagine whatever underlying code that implements the new AirPlay features. Tivo simply needs to make their app iOS 10 compatible, that is if they are still supporting the iOS app.
I'm not sure if this specifically works for the TiVo app, but for other apps you can AirPlay just the video/audio in iOS 10 by swiping up to open the control center, swiping left within Control Center to get to the media playback controls, and then selecting the AirPlay target in the dropdown underneath the playback controls.