Some videos from YouTube can neither be played on my iPhone nor on the Apple TV
When videos are recorded/saved, or in this case, uploaded to a video playback provider (youtube, vimeo, etc) the video is compressed into a code format for the computer. That compressed video is then decompressed on playback.

There are many different video compression formats out there, which is where the issue arises. Most mobile phones support less VCFs than a computer/computer browser does, so compatibility is the problem. 

Every mobile phone supports H.264 (MPEG-4) which is saved as a .M4A extension, so that's typically what you'd want to save a video edit as. There are other VCFs that will work, but MPEG-4 is the most reliable

Unfortunately, videos that aren't compressed into a format readable by a mobile OS will not be viewable. The only solution is if the video is recompressed into a new format and re-uploaded.