I only had about 300 characters left and wanted to pre-empt a couple counter-arguments to this theory.
First on Tyrion. People often point out that Tywin repeatedly states that Tyrion is not his son. Let's take a look at the last words of the last conversation Jaime has with Tywin:
"You are not my son." Lord Tywin turned his face away. "You say you are the Lord Commander of the Kingsguard, and only that. Very well, ser. Go do your duty."
Also, when Tyrion responds to Tywin's final proclamation of:
"You... you are no... no son of mine."
"Now that's where you're wrong, Father. Why, I believe I'm you writ small.
When he said that I got goosebumps because so much of Tywin's disdain of Tyrion's behavior we find out is also his own behavior, hypocrisy at its finest.
Also, just to add a little weight here's George on why Tywin denies Tyrion is his son (credit /u/bookshelfstud from this thread):
Simple answer: Lord Tywin is in denial. If he admitted that Jaime wasn't heir [to Casterly Rock], he'd have to admit that Tyrion was. And that he has a hard time swallowing.
It's not that he knew Tyrion was Aerys', it's that after having two beautiful golden haired children he can't accept that he created the monstrosity (in his eyes) that killed the love of his life and who now stands to inherit his legacy.
GRRM has outright stated in an interview (credit /u/roadsiderose from this thread the other day):
...[K]eep in mind that The World shouldn't necessarily be taken as gospel. The book is written from the viewpoint of a maester at the Citadel, one who hopes to pass its knowledge on to someone sitting on the Iron Throne. As such, the author may have ... rearranged events to suit the interests of a particular royal family. "So who knows if it's really true or not!" Martin chuckled. Furthermore, the maester's knowledge comes from other scrolls that, in turn, may be unreliable.
As he says, it is being passed to Tommen and no doubt Yandel has heard rumors that Tommen may be a child of incest. Regardless you don't want to paint the Lannisters or Baratheons in a bad light and piss off the King (or Queen regent). Lastly, much of the information we get on Tywin (such as the scroll discrediting the "scurrilous rumour" of Aerys and Joanna having an affair) is from Grand Maester Pycelle, an unashamed Tywin crony. There is no way if he even did have knowledge of an affair occurring that he would say so. In fact he would probably discredit the rumor to preserve Tywin's honor, which is precisely what he does.
I do not believe there is direct evidence yet for this theory any more than there is for R+L=J and we will not get the piece to confirm it until it is revealed in the books. In the mean time there's a bunch of circumstantial evidence which I mean to compile in this two-part post.
Couldn't find a neat place to fit this (or any extra character space):
Jaime ignored that. "If these flames spread beyond the tower, you may end up burning down the castle whether you mean to or not. Wildfire is treacherous."
"Lord Hallyne has assured me that his pyromancers can control the fire"... "Let all of King’s Landing see the flames. It will be a lesson to our enemies."
"Now you sound like Aerys." -Cersei III, AFFC
That's right, Jaime is directly reminded of Aerys by Cersei's actions.
The green light of the wildfire had bathed the face of the watchers... Even in the baleful glow, Cersei had been beautiful to look upon... She is crying, Jaime had realized, but whether it was from grief or ecstasy he could not have said.
The sight had filled him with disquiet, reminding him of Aerys Targaryen and the way a burning would arouse him. -Jaime II, AFFC
His sister liked to think of herself as Lord Tywin with teats, but she was wrong. Their father had been as relentless and implacable as a glacier, where Cersei was all wildfire, especially when thwarted. -Jaime II, AFFC
Honestly I feel like putting 3 more quotes just from Jaime II on here but I've already been a huge windbag so I'll get off my soapbox and let you fine folks discuss the matter.