Very old thread, but since it has been resurrected, I'll comment.
Although this capability may not be built into MusicBee, it is already available for free and is very simple to do. Just download Audacity (free and excellent audio recording and editing software). Audacity can capture anything that is being played on the same computer (although there may be some limitations in certain OSs and with certain audio cards). It makes no difference what application is being used to play the sound, since the capture is done through speaker loopback or the sound card Stereo Mix function. Once the file is recorded, you can edit it in any way you like, then export it in just about any format.
The operation is virtually identical to recording from LP or tape, except you do not need any external audio input. I export my audio tracks directly into the MusicBee library, and the file structure and identification tags are handled automatically.
If you want more information, start a thread in the General Discussion forum and we can talk about it there.