Sorry- I thought someone had actually anserwed it, my bad. Anyway, start the game and once you're actually playing, pause and hit F2 twice to get to the options menu (or I think it's the 'Under The Hood' menu) and go to audio options. Turn music volume to zero. Then hit Alt+Tab to close Burnout to the taskbar. Now you can open Windows Media Player or WinAmp or whatever to play music. Click the Burnout tab in the taskbar to re-open it and your music will now play while you drive around.
lol I thought this the first time I read the question, but its actually the official recommendation. And when you think about it, it isnt that sarcastic. Not every pc game minimizes without crashing, and the fact that this one does is something of a feature. It would be nice to integrate a music player (or at least an intelligent mixer) to allow fading of music while DJ Attomicon is talking, but it still is good enough.
100% British Beef http://files.deviantart.com/f/2004/137/4/0/BigRedButton.swf
I used to have GTA 3 for PC (all the others on PS2) and there was a custom music station that you could drop your own mp3s into a specific folder, and you could change stations. EA shoves their EA Trax down our throats in all their games- Madden, NHL, even Tiger Woods, and that's all we get from them. A bit bogus, but I either play with no music or run WMP under it. Not sure what the console situation is...