I have invitation alerts audio switched on. It only seems to work for friend invitations when I am actually playing a game.
Even if I have the game client OPEN, if I'm not actually playing a game, there is no tone.
Honestly, I wish I could turn off all the tones that occur as your mouse goes across a link. And if your mouse drifts over a link while you are doing something else, it will beep incessantly at you until you move the mouse.