You can disable sound in the Sound option, alt + o > Sound
This enables/disables sound events, not all sounds in mIRC, eg. using /splay, so this would not have an effect. Wims pointed out that this does in fact disable all sounds but has no effect on the issue being discussed above.
mIRC often needs to play multiple sounds one after another, eg. sound events, and repeatedly initializing and closing the sound card results in chopped/unplayed sounds, so newer versions of mIRC keep an open sound channel to ensure that sounds are played smoothly.
I might be able to change this so that the sound channel is enabled/disabled depending on whether Sounds/Do not disturb are enabled/disabled. I have added this to my to-do list.