If you shoot digitally at 23.976 fps, you still transfer to 35mm film at a 1:1 ratio, i.e. one frame of digital is recorded to one frame of film. So basically you are just projecting 23.976 fps footage slightly faster at 24 fps in the film print.
If you shoot digitally at 23.976 fps, you should record sound at that rate though it may involve selecting 29.97 fps on the sound recorder for TC (time code) purposes. This info is old but probably still relevant:
http://www.locationsound.com/proaudio/ls/tips/techtips17.html
http://www.petergray.org/doublesystemsound.html
If you shoot film, you can shoot at 24 fps and record the sound using 30 fps TC... since the footage gets telecine-transferred in 60 Hz countries at 23.976 fps (because both NTSC and HD work at 59.94 Hz instead of 60 Hz), the post house will pulldown the sound to match for editorial / dailies syncing purposes.
All of this is really an audio sync issue, not so much an image issue -- the trouble is that post is sometimes done at 23.976 fps or 59.94 Hz, but the final film-out is still projected at 24 fps, so at some point, the audio has to be adjusted to match if it doesn't already. So if you shoot at 24 fps and run the sound at the same speed, when you finally do a mix using a film print, your sound sync is correct, but while in a 59.94 Hz post, you were working with pulled-down sound. Whereas if you shot at 23.976 on a digital camera and the sound was recorded at that speed, and did the sound mix at 23.976 fps using a digital copy, then after the film-out, the sound sync has to be adjusted for 24 fps.
I'm not really an editor or a sound mixer so you'd get better advice on these issues from one of those types of people.
