After much investigation and several chats and calls to AppleCare, I've found out exactly what's going on, but I doubt that there will be a convenient fix forthcoming.
First, let me say that it doesn't involve simply waiting for the migration to Photos to finish. You can wait until the cows come home and these videos aren't going to play (I tried a clean migration several times, welcomed many cows to my home, and still no luck).
The videos that will not play in Photos are those that have Microsoft-compatible MP4 content. As has been discussed, these also will not play in QuickTime 10 (QT10 is used by Photos to display video). They're typically produced by several digital cameras - mine were shot on a Casio EX-Z850. They would play in previous versions of QuickTime (e.g., QT7), and in iPhoto, if you had a QuickTime-callable codec/plugin installed on your system that could play them. The most commonly used such codec is Flip4Mac (which recently became considerably less friendly - and a free version appears to be no longer available - but that's a bit unrelated). If you've been able to play videos in iPhoto and QT7 that will not play in Photos and QT10, take a look in your System Preferences; I'm pretty sure that you'll find Flip4Mac there. Or at least somewhere on your Mac.
The problem is that, for reasons I don't fully understand, QT10 will not use the Flip4Mac plugin (it may have to do with proprietary Microsoft code or licensing), and so Photos will not use it, either. If it involves licensing, it's very unlikely that Apple will start paying Microsoft for this, and that means that neither Photos nor QT10 will ever play these videos (at least not without a lot of mucking around on your part if it's possible at all).
The only reasonable solution that I've seen discussed is to convert the problem videos to a different codec that QT10 will play. There are a number of converters out there, including QT7, which can export video in various formats. Some folks have recommended VLC, which will also do the job. I'm sure that there are others. An inconvenience with this approach is that the creation time on your videos gets messed up. This can be fixed in Photos, but it's a nuisance. You'd ideally like these conversions to preserve everything but the encoding, but they don't (at least I can't find one that does).
This solution is okay if you only have a few such videos, but if you have dozens or hundreds, you'll have a lot of work to do. And it will take a very long time no matter what you've got for hardware. If you're pretty clever with AppleScript and/or other such automation tools, you may be able to automate the conversion so that at least you're not sitting at a computer for two months without sleeping. But it's not going to be fun no matter what.
There's one detail that I haven't yet looked into, and which somebody else might take a look at to see if it offers any help: the latest version of the Flip4Mac plugin is apparently no longer free. I haven't tried buying it to see if QT10 can now use it, but if you're brave and/or rich (mostly brave - it's not expensive) and decide to try it, please let us know if you can play your videos with QT10 after installing it. Unfortunately, there doesn't seem to be any trial period that would let you find out before buying. Personally, I doubt that even the paid version will work, so I don't want to waste my time and money for only a handful of videos. But if anyone can report back here that it's now working, please do!
