This one is so bad—but also more rare—I'm not even going to bother looking up what any grammar "experts" have to say about the mistake. It's just straight up wrong. I hear it occasionally, and I just heard someone at work say it last week.
Mute means silent, and nobody has any excuse for not knowing what it means. Everyone sees the word mute in writing every single day on their TV remote, and should directly understand what it means.
Moot is usually used in the context of a discussion, argument or debate. Here are Dictionary.com's top two results for moot:
1. open to discussion or debate; debatable; doubtful: "a moot point."
2. of little or no practical value or meaning; purely academic.
It usually goes a little something like this:
Two or more people are debating some mostly irrelevant point, when someone in the group pipes up and says, "It doesn't matter anyway. It's a moot point."
It is not, however, a mute point.