Negative impact or not, unlike job interviews, there is no common etiquette about sending personal notes to the audition panel. IMO, that means you should be careful in doing things that might be construed as presumptuous. (For instance, would you consider it appropriate to send a gift after an interview? Probably not.)
Personally, I’d let the kids make the call on this one. Some auditions are clearly much more personal than others, and it’s impossible to judge unless you’re in the room. My son has had an audition where the school representative did not introduce himself, and was basically a monitor who video-taped the audition. He’s also had an audition where the department chair took him aside afterwards, introduced himself, gave him his card and asked him to keep in touch. The same rule about post-audition notes may not apply to the two situations.
That said, the very impersonal audition resulted in an acceptance with half-tuition merit, so I definitely agree with those who advise not reading too much into the response you get at the audition. Also keep in mind that those making the final decision may not be the ones who were in the room.
A “thank you” is not part of the custom of auditioning in music. In a job interview, it is. In an audition, it is not. They are not the same thing. It’s good to know the customs for music auditions.
To share an inside perspective at a big music school, my D helped her teacher for a year with administrative stuff. Her teacher was well-known, still traveled a fair amount, had a full teaching schedule and a studio to manage. She also rarely took undergrads but still sat on all panels. My D shared that she would get annoyed at audition time with the extra work load. She would particularly get annoyed with the crush of emails. She would have my D screen as many emails as possible (and they were a lot). It would be nice to think that professors have time to really read, absorb and remember each email. But this is a very busy time of the year for teachers. Auditioning is in ADDITION to their full schedules. If you have no connection with some of the teachers on the panel, I would not do it. This experience really stuck with my D and she is very conscientious of the panel’s time and follows protocol. That is always appreciated!
As pointed out before, a thank you for sample lesson or if you have some sort of personal contact with a teacher from the past on the panel, then it is appropriate. Or if a teacher were to run into you in a corridor and say how much they liked your audition and spoke with you about your interest in the school then you could do an email. But the random teacher sitting in the room, not necessary.
This: PROTOCOL
Do not underestimate the importance of protocol. In music there are long standing traditions and ways of behaving that seem to not make any sense. Stupid stuff you need to learn along the way such as --singers do not wear open toed shoes and never ever wear purple at a performance in Germany. [-X
The panel will probably just mark it down to naïveté at best or at worst be annoyed. But why do it when you know better?
Besides, as my D said — if you truly have the goods, they should be sending YOU a thank you note.
This is why I love this place… learn something new everyday. Is there any rhyme or reason behind these?
" Is there any rhyme or reason behind these?"
Yes, but those rhymes and reasons probably date back to the 18th century. (fyi in Germany purple is a death color—or at least it was a few hundred years back)
D and I laugh about protocol all the time. We thought of a new one (for professionals only)—on your way out of the audition room, hand each panelist a $25 gift card to Red Lobster ( along with a wink). THAT would make an impression for sure. 
*****
It’s a long haul, just follow the rules ( but keep your sense of humor about it).
**** BTW ---- it’s a JOKE> don’t try this at home.
Protocol is so important. There is a certain dress, a way to walk into the room, a way to work with the accompanist, the right pieces to sing…it all shows you know the rules of the game. Also there is right and wrong in communicating with the teachers prior, during and after. And trust me, I’ve had a few moments of …Gosh darn it all, I’m the one paying all the money for her to audition here…you’d think they could be more polite, inviting and communicative! But they are following protocol…and it’s time to start getting used to it. It is something my D learned well in undergrad.
BUT I never want young people (kids entering undergrad programs) to freak out too much about protocol. It can tie you up in knots when you really should be focused on your pieces. Undergrads are NOT expected to know it all (and talent will trump protocol at this age). They should know some basics…like dressing nicely (even open toe shoes would be ignored), having their music in their binder correctly and neatly. Being polite to the accompanist and the panel. And being respectful of and grateful for a teacher’s individual attention any time in the process. Really common sense stuff. You have undergrad to learn and practice protocol. Still when you learn protocol here early, it only helps to give a positive impression.
Don’t wear purple in Germany? This is a thing?? 
D lives and performs in Germany. For classical vocalists it is a thing. So odd. :-??
Granted–this is professional protocol, but I seem to hear something new and bizarre every other week from her. In fact whenever she auditions or performs in another country she networks to find out the local do’s and don’ts .
This reminds me. I need to find out what the French equivalent to Red Lobster is.
8-}
La Langoustine Rouge of course.
Don’t wear yellow in Spain, for the same reason!
Not as many such chains in France. Buffalo Bills is a steakhouse with outlets around the country 