Can you create a viable two player turn-based networking implementation using only REST API for outbound and push for inbound data? If a game is comprised of, on average, 30 moves, is it a bad idea to just send 15 push notifications to each user? If hosting expense is less than income for each DAU, is this perfectly acceptable, or an abuse of what push is intended for? ...Why or why not?
Push Notifications are not ideal for this purpose as push notifications may come at long unpredictable delays, may come out of order, or may not come at all. Push notifications also encounter rate limiting at volume. In my comment to your question I said "Trust push like you trust UDP packets" that's giving push far too much credit. In most cases you'll find that sending push notifications comes with more overhead than just talking to your game client via your API.
You should probably just poll your game API from the client when the user is currently in game. If a user has not made their move and it has been some time, that may be a good opportunity to send a push.
This answer is somewhat predicated on the assumption that by turn based, you mean a pace similar to games like Hearthstone. If your game involves turns that are likely to be slow paced like a play by email game then push may not be a bad option, but it won't be a reliable option in the same way that email is.
An additional note, Apple Push Notifications for iOS are typically much less reliable in practice than Google's new Firebase Cloud Messaging for Android (Google Cloud Messages, their previous service seemed to be about the same in my experience).