Try uninstalling the Iplayer app then installing the bbc media player app first.
Then reinstall the Iplayer app, it does work with both sd and hd versions of the fire.
There also is a known issue with some wifi connections which the BBC have said will be fixed in the next update.
If you find it won't connect try changing the stream quality in the settings .
This is the problem the BBC are trying to fix,I have a 20Mb broadband connection and have this problem when the app is set to high quality streaming.But works at medium and low.