The idea that the larynx should never move and that, in fact, nothing should move while singing is simply silly.
When we are singing everything that can move should move and move freely. The movements should be smooth, coordinated and not unpleasant to do or to watch. Keeping everything absolutely still will cut off expression, make it impossible to change the quality of the vowel and deaden the vocal production so that the voice becomes uninteresting or even unpleasant.
Why would anyone teach such an idea? Why, indeed? This, and many other ideas like it, are out there and sincere students seeking to learn how to sing run into them every day. In fact, doing anything deliberate with the larynx is a bad idea, as the larynx isn’t something we should be trying to feel, let alone move. And, if you can’t change your jaw position, you can’t change the formants very much. If you can’t move your face muscles, you can’t change the vowels much, and that, too, limits formant “adjustments”. If you can’t allow the larynx to adjust (indirectly) you will have a terrible time singing high notes or soft tones. Bad ideas every one.
The larynx is a joint. What joint in the body does better when it can’t move? Smooth movement of a joint depends on flexible muscles and coordination. Muscles develop flexibility through stretching, but in order to stretch them you have to move them — to pull on them repeatedly. What happens when you deliberately don’t move? They get stuck. They atrophy. Then it gets nearly impossible for them to do their job in supporting the joint to carry out its primary function, whatever it may be. Trying to “unlearn” that is quite difficult, especially if you do not have a teacher who knows how to give you help.
Attempting to control the things you should not be trying to control and not controlling the things that need to be calibrated deliberately is like standing vocal training on its head. This is where the inexperienced teachers have no clue about what is cause (the vocal folds vibrating) and what is effect (the vowels we hear as acoustic response) and the purpose of airflow (moving across the vocal folds as they vibrate). You can’t teach at a high level if you are busy telling people to do things that (a) are not under direct control, (b) are best left alone.
How many times over the years have I had someone come to me for lessons who has been taught and who has deliberately practiced behavior that was detrimental to both the voice and the heart. Singing training like that closes you up, cuts you off from your body and makes you sound like a machine.
If you are studying with someone who tells you not to move, you need to ask why. You need to LISTEN to your teacher sing, throughout his or her range, and LISTEN to how they sound overall. You need to look at his or her face and body while they are singing. If it looks awful and sounds pretty bad, change teachers.