How to save settings in esc menu?
Solution 1:
At this moment of writing, Game Dev Tycoon is very, very bogged down with miniature bugs like that. It's supposed to save changes in the settings by itself when you exit the settings menu, but it doesn't always do that. For me, muting the music is only a temporary thing, with the music returning again after about 30 minutes on its own, the "Music" checkbox still unchecked in the settings menu.
I am unsure how it is now, but you can't mute the game in the Windows Volume Mixer either, because its icon is too big for the volume mixer window so that the slider itself is outside of the window, rendering you unable to mute the game.
In short: it's a bug, so no. Not at the moment.