Yes, finally (and vice versa)!
In the aftermath of Apple’s announcements earlier today, it released an update to iTunes with a new feature described as rent once, watch anywhere. The release notes for iTunes 12.6, which is available as a free upgrade on the Mac App Store, say:
Now you can enjoy your iTunes movie rentals across your devices with iOS 10.3 or tvOS 10.2.
With this new feature, you should be able to start a movie rental on an iOS device and finish it at home on an Apple TV for example, which is a welcome change to what was an overly inflexible system.
I was finding myself in the situation of renting a movie on Apple TV and wanted to continue watching on another iOS device (mostly iPad) a few times, and even didn’t rent movies because of that restriction. Or I was browsing on iPhone what movie I could watch when at home and it would’ve been just nice if I was able to rent it straight away. Also useful to add content to a “watch later” list, e.g. for $.99 movies or documentaries that are only available at that price for a short period of time.