As an adjective, moot originally meant arguable or subject to debate. With this sense of moot, a moot point was something that was open to debate. But, since around 1900, the adjective has gradually come to mean of no importance or merely hypothetical. This usage arose out of an exercise in U.S. law schools involving the discussion of “moot” cases to practice argumentation.
In the common phrase moot point, moot means (1) of no importance or (2) merely hypothetical. This is where moot most often gets confused with the adjective mute, which means (1) refraining from making sound or (2) silent.
Moot also has a verb definition—to bring up for debate—that is almost nonexistent in American English and rare in British English.
Mute point occasionally works as a bad pun, but it’s almost always a misspelling. For example, these writers use mute where they presumably mean moot:
Manager Joe Maddon brought in left fielder Russ Canzler as an extra infielder to hold the Gload at third, but the left-handed-hitting Naughton lined to right to make it a mute point. [American Chronicle]
If the NDC plans not to use violence in 2012, this whole issue becomes a mute point. [Ghana Web]
Although this could be a mute point since the downtown group will have plenty of time to catch up. [NFL]