I had a similar problem where the stereo would only operate with the door open. Even with the keys out - open door hear music, close door silence. Drove home in silence pondering the situation.
Checked fuses, found the fuse for the vanity lights (which also covers the stereo and illumination control) blown. Come to find out junior touched both contacts with a penny while I was getting the little one from school.
Take a look at fuse #2.
'73 280SEL 4.5 (9/72)- RIP
Only 8,173 units built from 5/71 thru 11/72
'02 CLK320 Cabriolet - wifey's mid-life crisis
2012 VW Jetta Sportwagon TDI...at least its a diesel
Non illegitemae carborundum.