(this was mostly written on Friday 14th December, although no-one else seems to have stepped up to the plate yet)
There are some native, commercial, applications to do this on Windows like Replay Media Catcher but as far as I can tell, nothing on Linux. There has been a first-pass MythTV integration but this seems to use an embedded browser window.
Red5 is a Java-based open-source Flash server which also comes bundled with some code for a sample client (download the tarball and then look in src/org/red5/samples/client). It might be possible to use this to download the stream; I haven’t yet tested. There also appear to be some code samples in Ruby and Python lying around, but these look very incomplete.
The URL for the stream is along the lines of this:
where both SECRET_KEY and STREAM_NAME come from a file of the name http://www.bbc.co.uk/mediaselector/3/stream/check/iplayer?pid=PID where PID is a BBC Programme ID such as b008h3zq as found on http://www.bbc.co.uk/programmes/ – the base URL for this address is found in the config.xml file linked to from every iPlayer page: http://www.bbc.co.uk/iplayer/emp/xml/config.xml
So this looks hackable, with the most promising chance probably being with Java at the moment.