No, TuneIn radio only streams. Granted, it's an excellent app for listening to Internet radio, but it still is just for streaming.
I'm afraid there's no app for this - as any app that allows you record music from streaming from the Internet would be walking a fine line with piracy. Music streaming is for streaming, not downloading so you can burn it to a CD for playing elsewhere/else when. That's what CDs are for (or buying MP3s from iTunes or Amazon).
As this question is perilously close to asking how to get something (that should be purchased) for nothing, I'm going to close it so no one has to jump over that fine line I described. Thanks for your understanding.
Marilyn