When the latest app update was released, did you install it over the one on your device ? or did you close the app, completely un-install, then install the new app ?
Have you tried closing the app, completely un-installing and re-installing
I've tried it with a very strong wifi connection on my iPod and the videos are playing for me.
Issues with the chess.com iPhone and iPod update





Well, I can understand it's not a priority... but at least confirm the issue... and say you are looking into it...

It is being investigated. It makes it harder to replicate though when some have no problems at all and some do.

I tested this myself, since I own an iPod Touch 3rd Gen, with the latest operating system installed (iOSX 4.3.3).
I have the very latest chess.com app download (Version 1.5.2, June 13th, according to the app store).
While watching Part 2 of the Qd6 Scandinavian by Perelshteyn video, my screen went to the Quicktime screen a couple of times by itself, with the audio still playing (as described here and elsewhere). However, all I had to do was touch the iPod Touch screen and the video resumed. At that point (some 5 to 8 minutes in or so), the rest of the video played without any adverse effects.
So, I decided to try it all again. I went into a different room of the house (closer to my wifi connection). I opened up the video and started it all over. This time, the entire video played without interruption at all.
This sometimes does / sometimes doesn't effect might suggest that there's nothing inherently wrong with the app or the video, but might have to do with signal strength in various locations. At least that's one possible explanation - though I could not replicate this by watching another video is a distant location.
So ... I'm not sure what is going on, myself - but it doesn't appear to be a critical issue in my case.
I like the new updated chess.com for iPhone and iPod, but there is an issue with the playback of videos: On the iPhone 4 the videos play well until after a few minutes, when the picture disappear and a big QuickTime "Q" appears whilst the sound remains as the video plays on. With the iPod with the latest system update I can't play videos at all.... Has anybody else experienced the same?!