I haven’t done this with appliances but usually do with calculators that beep every time I press a key.I've done this on other appliances. The beeper probably looks something like this, although it might not be black:
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All you need to do is stuff something (I like the hot glue idea!) in or over the hole where the sound comes out. If you stuff something inside and prevent the diaphragm from moving you may be able to silence it completely, or nearly so. Just a piece of tape over the hole will quiet it down.
These things can be tiny. You're not looking for a car horn!