See subject line. As near as I can tell, “takes the piss” is a reference to some person or entity engaging in comedy.
What is the origin of that phrase, and can someone provide a more precise meaning than I’ve got there?
Either (1) taking liberties, or (2) making fun of something (more politely Taking the Mickey).
(1) Our MPs? Their expense claims were just taking the piss.
(2) On Top Gear they often take the piss out of the French*
Origin, hmm. Probably just a rude variant on taking the Mickey, and lawd knows where that came from.
- or the Germans, or Mexicans or Porsche drivers…
I concur with these two precise definitions. It isn’t just engaging in comedy as the OP suggests. In fact it doesn’t need to be comedic at all, it can be downright offensive. Eg:
“hey, has your [ very fat ] girlfriend taken shares in McDonald’s?”
“You taking the piss?”
Pub brawl ensues.
Also:
To take advantage or push your luck:
“hey boss, can I leave an hour early to pick my girlfriend up from her exams?”
“Sure!”
“Can I also come in late tomorrow? Might have a hangover from the celebrations”
“Don’t take the piss”.
Take the Mike out of shows up first in print[1935) before the other versions. The OED speculates that it might refer to Mike Bliss as rhyming slang.
Thus, Mike—Mickey.
I don’t think it means “taking liberties”. MP’s expense claims (and similar examples of this usage) are “taking the piss” because it is such barefaced affrontery that it is as if they are making fun of us, as if they are laughing at us for how easily they have fooled us. I think the expression always means to make fun of, to ridicule, in a belittling way. I have no idea of the origin either, however.
Perhaps it worth noting, however, that in British slang “to be pissed” means to be drunk, not (as in America) to be angry. (However, to “be pissed off” does, in Britain, mean to be angry or annoyed. When one is angry with someone, one is pissed off at them.)
Possibly, just as a WAG, taking the piss has to do with drunken ridicule.
Maybe it was once “Mike”, but in my lifetime it has always been “taking the Mickey” or “the Mick”, or possibly (humorously) “the Michael”. Who TF is, or was, “Mike Bliss”? I have never heard of any such person, and if he were not someone once very well known (at least in the East End of London) surely his name would not have been used for rhyming slang.
Anyway, if it is is rhyming slang based upon “Mike Bliss”, that would explain only how “taking the Mike” might derive from “taking the piss”, not where “taking the piss” itself comes from.
I have heard two competing theories for “taking the Mickey”, both based on “taking the piss”. One is the rhyming slang “Mickey Bliss”, which is given in various dictionaries (but nobody seems to know who he was). The other is that it is short for “micturition”, a medical term for urination, i.e. a faux-grand euphemism for a common slang term.
The latter idea seems a bit far-fetched, but then my grandmother also used a sort of medical variation on the term: she’d refer to “extracting the Michael”, or, if she was feeling especially whimsical, “performing a Michaelectomy”. 
I have no idea if that was original to her, though.
As for the origin of “taking the piss”, I believe it refers to the idea of puncturing someone’s pomposity, cf the phrase “all piss and wind”, meaning “full of noise and bluster but with very little substance”. If you take the piss out of someone who is all piss and wind, then there’s not a lot left.
See the discussion here, for instance.
In a similar vein, World Wide Words says that “piss” was slang for “pride”, so the phrase may have come from that, the sense being to deflate someone’s pride. As for the politer “Mickey” version, no idea. Maybe someone with a good turn of phrase just came up with it and it stuck? I’m not sure that idioms need any better explanation than that.
Mickey Bliss, not Mike. Nobody knows who Mickey Bliss was, though supposedly he was a BBC radio broadcaster in the early part of the century.
There’s an etymology, which I’ve no idea how accurate it is in that i refers to when human urine was used to tan leather and there would literally be people whose job it was to go around “taking the piss” in order to collect it for use in tanning.
Taking the piss means to make fun of somebody or something. It can also mean to take liberties, in the sense that if someone was taking liberties (e.g. eyeballing your pint), one might ask them “you taking the piss geez?” as in “are you serious sir?”
Agree.
As for origin, I have no idea either but my WAG is it is to do with Chamber pots back in the day …