Professional TV and radio shows never play music while someone is being interviewed, and you guys shouldn't either.
- Peregrine Falcon
That entirely depends on whether it is a serious sitdown interview or a feature. I've worked in radio a little over 20 years and have produced countless examples of what you're describing. Sometimes there is no music, like if it's a segment to be played back later as if it is happening live. Other times, an interview is cut much shorter and couched within the framework of a featurette -- something like a four or five minute segment where the answers are heavily edited into soundbytes. For this I'll often have a logo at the beginning and then separate music underneath throughout. They music has to be kept low enough and be non intrusive enough so that it won't take away from what's being said. I usually use some generic electronica from a production library as a bed.