I found that too - it's incredibly stupid in my opinion, does anyone know why they do it?
In the case of 4oD, it's due to licensing agreements with their content providers. According to a post regarding the matter on the Channel 4 website that I recently read, it's something they want included in all future agreements.
As others have reported, the BBC iPlayer app works over AirPlay if you are streaming. I have found, however, that if you download a programme to the iPlayer app, and then try to play it over AirPlay, it does not work. If you use AirPlay mirroring, then it does.
The ITV app only streams audio over AirPlay, but works splendidly with AirPlay mirroring.
4oD simply does not work on AirPlay for the aforementioned reason.
When I got my Apple TV 2 two years ago, it seemed obvious that an iPlayer app, at the very least, would quickly be added. Now it seems less likely. Of course the AirPlay feature overcomes the problem to an extent, but it seems like a lot of knobbing around for a feature that seems to built-in to every other electronic device sold in the UK. I need to buy a dehumidifier (recommendations welcome!), and I'm half expecting that to feature an iPlayer app.