Will Apple TV work as a DVR with Cable TV apps?

Apps from TV providers may have “Cloud DVR” options (for live TV), if you subscribe to that (sometimes an add-on service). ‘Recorded’ programming will then be available from a user-specific area in their app. It is up to the app developer to include that functionality in the app.

Cloud DVR from internet streaming TV providers:

Manage Cloud DVR recordings - Hulu Help Center

Record shows, sports, events, and movies - YouTube TV Help

Record Live TV with Cloud DVR - Sling TV Help

Record and watch shows with Cloud DVR - ATT TV Support

How does Cloud DVR on fuboTV work? - fuboTV Help Center

Using Philo - Saved shows (DVR) - Philo Help Center

Apple TV's are streaming devices and will not record content. There is no reason to as content is available to watch anytime as long as the provider continues to provide the content.