When a woodwind or brass instrument is played, how much air, if any, comes out of the tone holes? Or does the air go into the tone holes due to the Bernoulli / venturi effect?
Modern brass instruments do not have "sound holes" -- all of the air blown into the horn exits through the bell of the instrument.
On woodwinds, air comes out both the sound holes and the bell. How much air vents through the holes depends on how many holes you have open. The more open holes, the less air will exit through the bell, and the more through the holes.
You can easily verify this for yourself. Take a clarinet or a flute and stuff a sock in the end. You will still be able to play all but the lowest note, and while the lower notes may sound a bit stuffy, you may not even be able to hear a difference in the higher notes.
This has ramifications for amplifying wind instruments, since not only air, but also -sound- emerges from the holes. A lot of people mistakenly point a microphone at the bell of a sax or clarinet, which makes the lowest notes "honky" and the upper noted weak. In fact the mike should be pointed at about the center of the instrument body, to pick up the sound emerging from the holes, and allow better balance over the full range of the instrument.
I would say that the amount that comes out the sound holes and the bell is directly related to the amount and pressure of the air that is being put into the instrument at the mouthpiece of the instrument. Internal pressure in an air column such as a sax is greatest at the mouthpiece and decreases as the air column approaches the bell of the instrument, so the air escaping through the sound holes near the mouthpiece does so at different pressure than the sound holes that are located closer to the horns bell. Also the sizes of the different sound holes would also affect the amount of air escaping through the hole.