The "the liberties [Aerys] took during the bedding" as the basis of a theory for Jaime and Cersei doesn't fly according to information from the recently released book The World Of Ice And Fire, as quoted by Tumblr user joannalannister:
The theory “Jaime and Cersei are Secret Targs” was based on the idea
that Cersei and Jaime were conceived when Aerys allegedly raped Joanna
while taking “liberties” at the bedding on Tywin and Joanna’s wedding
night in 263 AC.
[...]
But even if Aerys HAD raped Joanna while taking “liberties” in 263,
it’s not physically possible for Jaime and Cersei to have been
conceived on Tywin’s wedding night in 263. Jaime and Cersei were born
in 266 AC at Casterly Rock. Joanna was not at court; she had been
dismissed immediately after “liberties” because Rhaella thought of her
as a “whore” and victim-blamed her. (Click here for relevant
quotes.) Joanna was not in contact with Aerys again until they
moved the court to Casterly Rock for a year in 267 AC, which was after
Jaime and Cersei were born.
UPDATE: Tumblr user joannalannister writes this regarding Tyrion being a Targaryen (as the result of Joanna Lannister having been raped by Aerys):
I am certain Tywin would not continue to serve the man who raped his
wife, regardless of what Aerys wanted. Tywin resigned over Jaime being
taken from him; he would have resigned over Joanna being raped. I feel
very confident in saying that.
There is a lot more in her post, though it should be noted that there currently is no evidence to definitely rule anything out. But to me it does seem to be a case of GRRM trolling his readers; like she says:
Saving the world is not limited to Targaryens, secret or otherwise.
UPDATE 2: It appears influential ASoIaF blogger Sean T. Collins is now on board with "Tyrion Targaryen" after reading The World of Ice and Fire. And he doesn't seem to be trolling his readers.
I've just started reading TWoIaF so I don't know what "evidence" he found; I'll try to update my answer once I reach that part of the book.)
UPDATE 3: Elio M. García Jr. and Linda Antonsson, co-authors of The World of Ice and Fire, spoke in the Podcast of Ice and Fire about this topic:
They point out that the new book pretty much flat-out debunks the idea
that Aerys is secretly Jaime & Cersei’s real father. That’s a theory I
never thought cut any ice anyway, but they’re right, the timeline
doesn’t work in any way. With a surprisingly vocal degree of regret,
however, they also point out that it doesn’t debunk the idea that
Aerys is secretly Tyrion's father. Interestingly, they're on the side
of the many, many detractors of this theory here on tumblr — they
don't like it at all. Elio said something to the effect of “You know,
it's George's story, if that's where he wants to take it that's fine,
but…” and he and Linda went on to voice some of the same objections
I've seen here, off the top of my head the idea that it undermines the
weight of the similarities between Tywin and Tyrion, and that this
might be one secret Targaryen/Blackfyre too many. The whole
conversation is interesting insofar as if they'd had any inside
information about the truth of the matter, they surely wouldn't have
brought it up at all, and they dislike the idea enough to say so
publicly even though by their own admission it's entirely possible
their own co-author is headed in exactly that direction.
In conclusion: most likely we'll only know for certain after the publication of the final book in the A Song of Ice and Fire series.
UPDATE 4: I've read the relevant passages in TWoIaF, and it seems to me there is indeed no reason to assume Cersei and Jaime are Targaryens. However, Tyrion's origins seem to be deliberately vague. There is no report of Aerys II Targaryen bedding Joanna Lannister, but he had lusted after her even though he seemingly hadn't consumed that relationship before her marriage to Tywin.
I see it entirely possible that he abused her at the Anniversary Tourney of 272 AC, held to commemorate Aerys’s tenth year upon the Iron Throne, especially considering his foul remarks towards her (he asked her if giving suck to them had “ruined your breasts, which were so high and proud.”). Tywin attempted to return his chain of office the next morning, but the king refused to accept his resignation.
There are no "exact dates", but I think it is a bit too coincidental that she gave birth to Tyrion in 273 AC and that some of the descriptions match the descriptions of the miscarriages Aerys's wife suffered:
a malformed, dwarfish babe born with stunted legs, an oversized head,
and mismatched, demonic eyes (some reports also suggested he had a
tail, which was lopped off at his lord father’s command)
Then there's also Aerys's jealousy WRT Tywin having beautiful twins:
“I appear to have married the wrong woman.”
There's his insistence that Tywin bring them and their mother to court:
“And bring their mother, too, for it has been too long since I gazed
upon that fair face.”
And there's his pronouncement after the birth of Tyrion:
“The gods cannot abide such arrogance. They have plucked a fair flower
from his hand and given him a monster in her place, to teach him some
humility at last.”
None of that is evidence, of course.