It looks like the current version of Air Video HD doesn't work with the lightning digital av adapter. At least, the newest rev of the adapter... I don't have experience with the older hdmi adapter. The newest is MD826AM/A, which replaced MD826ZM/A a few months back apparently.
Symptom is it will display "Playing on HDMI" in the bottom but the playback never really starts, instead it starts alternating between the frame of video and black on the tv. Almost like it is getting confused on where it is suppose to be playing back Air Video HD interface locks up, can't pause etc. And mirroring sometimes doesn't come back after quiting + force quiting air video hd, have to remove and reinsert adapter.. I tried this on two TVs, two different manufacturers. Netflixs and display mirror work fine, so the adapter itself seems to work (I'm starting to wonder if it is really just a conflict between airplay mirror and the straight airplay video... I wish they had an option to shutoff airplay mirroring on the lighting adapter).
This is on an iPhone 6S Plus and an iPad Air... same symptoms of both, and both running iOS 9.2.
I just discovered the same problem earlier this week. I was at a hotel, and I attempted to connect my iPad to the HDMI connection on the TV, and experienced the same behavior that bwb described above (and bwb didn't mention this, but in addition to the picture issues, I don't get any audio either). It looks like the HDMI handshaking may not be working properly, or something along those lines - basically, it appears that things are not syncing up properly. I returned from my trip last night, and tried it on my TV here at home, and have the same problem.
A little more info. I have several other IOS apps that can play video - GoodReader, oPlayerHD Lite, PlayerXtreme Media Player, and VLC. After I found the problem with Air Video, I tried these other apps, and all four were able to play video through the HDMI adapter with no issues. So this problem seems to be specific to Air Video. Also, I stayed at the same hotel about a month ago, and HDMI output with Air Video was working fine at that time (using the same iPad, same lightning to HDMI adapter, same HDMI cable, etc.). This leads me to suspect that this feature broke somehow as a result of the upgrade to iOS 9.2, because I think the iOS upgrade is the only change in the last month.