The phrase owes its origins to the terms "Piss proud" and the command "piss off".

"Piss Proud" is defined by Francis Grose* as "Having a false erection" (from pressure in the bladder). This phrase was used (possibly later) to define one who would 'over inflate' their importance or actions, leading to the instruction to "piss off", effectively a statement that the listeners didn't believe the tale and that the speaker should stop acting so self important.
From this we can easily equate "to take the piss out of someone" to be the action of reducing pride (pomposity pricked).
To give the phrase a faux politeness it is a short step to "take the micturition" (or "take the micturate"), in turn shortend to "Take the mickey".

At least that's the tale I got it

Walrus


* Dictionary of the Vulgar Tongue.