Apple has updated iTunes for iOS and tvOS to version 12.6, which lets you rent videos once and watch them on multiple screens. However, it requires iOS 10.3 and tvOS 10.2, both of which are in beta testing. I’d expect to see the finished versions arrive on Wednesday or Thursday.
With the “rent once, watch anywhere” capability, you can rent a movie on one device (iPhone, iPad or Apple TV), then finish watching it on another (iPhone, iPad or Apple TV). Prior to iTunes 12.6 you had to finish watching a film on the device from which you first rented it.
However, Apple did allow movies rented on a Mac or PC to be transferred to an iOS device if you synced via USB; however, that content would disappear from your computer’s iTunes library. Of course, you could (and can) stream downloaded rentals from iOS and macOS devices to an Apple TV with Apple's AirPlay.
To get iTunes 12.6, launch the App Store app and click on the Updates tab at the top. You’ll see the iTunes 12.6 update, which is a free download.