Moot point is correct. Mute point is wrong, although a fairly common malapropism among morans. The less common "mood point" is also wrong.
"Moot" is a very old word related to "meeting", specifically a meeting where serious matters are discussed. The word moot derives etymologically from the Old English word gemot, denoting "meeting". 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.
The "mute" spelling is a development that has come about because moot is now a fossil word, usually encountered only in the phrase "moot point". There is an understandable tendency to convert the unknown into the known, and mute seems to fit the new meaning rather better. But it's wrong.
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