When you update to iOS 10.2, the old Videos app is deleted and replaced by the TV app. If you try to sync videos from a computer using iTunes — such as a movie or TV show — they are supposed to show up in the TV app in the Library tab at the bottom. But as I was transferring some videos from my computer to my old iPhone (which my son now uses without a SIM card), whenever I tapped the Library tab in the TV app the screen said "No Videos." That made no sense because I knew that I had indeed transferred the videos. For example, if I went to Settings -> General -> Storage & iCloud Usage -> Manage Storage (under "Storage") -> TV, each of the movies that I had downloaded to the iPhone were listed. They were on the iPhone, and were taking up space on the iPhone, but did not show up in the TV app and thus could not be played.
I found the solution after digging around for a while on Apple's online support discussions. It turns out that in iOS 10.2, you need to install something that you bought from iTunes before you can watch something that you transferred from a computer. This is obviously a bug, but there is a work-around that you can use until Apple fixes this. If you go to the bottom of the TV app and tap on the Store icon, you can scroll to the bottom to see buttons to press that will let you buy content from iTunes. I recommend that you tap Free Episodes. That opens up the iTunes app, and if you download anything — including something that is free — the TV app works. For example, I downloaded a very short trailer under the "Free Featurettes" section. After you buy or get something, go back to the TV app and now you should be able to tap the Library button at the bottom of the screen and you will see all of your transferred content.
Finally, I noticed that after I added more movie content to the iPhone, I would sometimes have to force quit the TV app and launch it again to see the additional downloaded content.
Hopefully this fix works for others too. And if anyone finds a better work-around, please post about it in the comments. I assume this bug will be fixed in the next iOS update, but who knows how long that will take to come out.