Failing to Connect Remote App to AppleTV (4th Gen), While Screen Sharing Works
Screen sharing and AirPlay can work over Bluetooth -- WiFi is not necessary. On the other hand the Remote app requires that the AppleTV is connected to WiFi.

This explained my problem.  My AppleTV was connected to a 2.4 GHz WiFi network, which stopped working. For some reason the 5 GHz network continued to work and my iPhone switched over, so I didn't notice.

Since I didn't have an original remote, I was able to follow these steps to switch the AppleTV over to the 5 GHz network:


Find a regular TV remote (with directional pad buttons). 
Plug the ethernet cable to the Apple TV.
With the iOS Remote App go to Settings -&gt; General -&gt; Remotes
Select Learn Remote and follow the steps to teach AppleTV your TV
remote. 
Once done, unplug the ethernet cable and set the WiFi under
General -&gt; Network -&gt; WiFi.