This is exactly what happened to me. I updated to IOS7 and my HDMI to iPhone and iPad cable worked fine with both devices. Yesterday at noon it was working fine. At 7:00 yesterday evening it stopped working. I had a message that my TV was not compatible and that if I was using AirPlay that I'd have to turn off mirroring. This was the same message both on iPhone and iPad. Very discouraging.
I've been using NetFlix exclusively for over a month, as when I got the HDMI cable from Apple, Amazon movies had a similar message.
I've determined that the change happened because under Settings/ iTunes & App Store/Automatic Downloads, I had updates on. Netflix Version 5 must have been downloaded and updated betweeen noon and 7:00 yesterday, and that is the problem. I then found that the older version of the NetFlix app was still on my Mac, V4.3 or so. I deleted the V5.0 from my iPhone, went to iTunes on my Mac, selected the iPhone, went to apps, and selected "update." After syncing, IT WORKS!