3 main approaches
1) You remove the trigger of the item in question.
2) You find a way to silence the effect. Many systems will have a software volume of some form (which most games for most of them I have ever seen will be left full volume or half on rare occasions) but looking at hardware here then anything will be software (and the SNES has its own audio processor setup so eh).
3) You break the sound somehow -- if it starts off as a microsecond of silence or whatever then you might be able to loop that back, or call it the end of the file immediately after that.
Most of those are "once you found it then it is easy" but the finding it part is potentially harder than you might care for in a first hack. For some later systems you will have an easier time but while this got ported a whole lot most of those will still be Japanese (and apparently the GBA version is a nightmare to hack).
1) After you press the button a whole chain of events will happen, one of those will be the sound. I don't know what we are suggesting for the SNES equivalent of https://tasvideos.github.io/fceux/web/help/fceux.html?TraceLogger.html
these days but such a thing should do here if you can find an equivalent (should be one but I am not so familiar with the current SNES debuggers). The idea will be sit at the last point on the menu before the sound is called, start it logging and allow it to experience just sitting there doing not a lot (any background audio, animations... will happen over a few seconds), now say what happens next is what I care about, press the confirm button and then what new stuff happens to be sending data off to the sound chip or if the sound chip reaches out and touches something you have some idea of what is going on, you then either prevent the call from ever happening or change the call to jump to the end of its own little bit).
2) Volume does not appear to be a thing in the articles I just read on the SNES audio ( https://mechafatnick.co.uk/2016/08/19/the-mysterious-legacy-of-the-snes-soundchip/#player1?catid=0&trackid=0 https://megacatstudios.com/blogs/retro-development/creating-music-and-sound-for-snes-games-a-crash-course-in-snes-gss https://en.wikibooks.org/wiki/Super_NES_Programming/SPC700_reference https://en.wikibooks.org/wiki/Super_NES_Programming/Loading_SPC700_programs
) so you get to find what generates it and convert that to silence. Might be a nice little wave file you can replace with silence.
3) That will see you want to understand the loop/function that runs that you might have found in 1) and alter accordingly. Technically it still plays but without getting too philosophical if a sound effect makes no sound did it ever play? Answer is probably yes if you care about memory and CPU but if you mainly only care about not having your teeth set on edge then hey.