I presume that it’s because it ruins the shot. Particularly shots where they’re shooting the car traveling towards camera (the camera being on a tow dolly) and the driver and front side passenger are having a conversation. I haven’t been on any shoots where this was done, but that’s my guess. Thelma and Louise had theirs, but jeez would that be a glaring mistake were it missing. 

Moviecliches.com has this to say: “Film cars do not have inside rear-view mirrors. Most of them do, however, have an appx 1” gray spot on the inside of the windshield where the mirror would normally mount.”