I back up the arguments of Dave Mortimer using the following words which you could find alone in the Internet.
“Moot” is a very old word related to “meeting,” specifically a meeting where serious matters are discussed. Oddly enough, a moot point can be a point worth discussing at a meeting (or in court)—an unresolved question—or it can be the opposite: a point already settled and not worth discussing further. At any rate, “mute point” is simply wrong, as is the less common “mood point.”
But what’s this about a “mute point”? As an acquaintance reports to me, some people say this thinking it means, “Let’s put the mute button on and cease any discussion on this.” There is an expression “to stand mute of malice”.
Wouldn’t it be funny if the term evolves this way to become correct? After all, with the ubiquity of remote controls and mute buttons, a “mute point” may make more sense than a “moot point” to someone who’s not a lawyer.
For today, however, it’s wrong. Say “moot point” and try not to stick a “y” sound in there.