Why does this potentiometer in an op-amp feedback path cause noise when adjusted?
The noise is caused by minute mechanical vibrations of the pot wipers on the rings (the latter are the resistive material.) Since neither the wiper nor the ring material are atomically smooth where they touch, rubbing the wiper on the ring produces slight vibrations. Some of that vibration is perpendicular to the contact patch between wiper and ring. The resistance of the wiper/ring contact varies with the normal force. This transient/AC variation of the contact resistance is what you're hearing.

How to fix it is (IMHO should be) a separate question. And non-trivial because adding caps across any pot terminals will change the filter characteristics.