After it was revealed that Oberyn poisoned his blade to ensure the death of the Mountain even if he lost, I read that there is a fan theory that Oberyn also poisoned Tywin before the duel. What evidence is there from the books that this may be the case?
There is no evidence that he did, but...
I suggest everyone to read the theory itself (also quoted below), and judge for himself. There are several passages in the books that allow for this interpretation.
Spoiler notice: Contains full quotes from the books.
Tywin Lannister: Dead Man Shitting?
“Where will I find my lord father?” “In the solar with Lord Tyrell and Prince Oberyn.” Mace Tyrell and the Red Viper breaking bread together? Strange and stranger.
—Jaime and Ser Meryn Trant, upon Jaime’s return to King’s Landing
“Widow’s blood, this one is called, for the color. A cruel potion. It shuts down a man’s bladder and bowels, until he drowns in his own poisons.”
—Grand Maester Pycelle, during Tyrion’s trial
“To be sure, I have much to thank your sister for. If not for her accusation at the feast, it might well be you judging me instead of me judging you.” The prince’s eyes were dark with amusement. “Who knows more of poison than the Red Viper of Dorne, after all?” […]
“Your father,” said Prince Oberyn, “may not live forever.” Something about the way he said it made the hairs on the back of Tyrion’s neck bristle. Suddenly he was mindful of Elia again, and all that Oberyn had said as they crossed the field of ashes. He wants the head that spoke the words, not just the hand that swung the sword. “It is not wise to speak such treasons in the Red Keep, my prince. The little birds are listening.” “Let them. Is it treason to say a man is mortal? Valar morghulis was how they said it in Valyria of old. All men must die. And the Doom came and proved it true.”
—Prince Oberyn Martell and Tyrion, in Tyrion’s cell
He found his father where he knew he’d find him, seated in the dimness of the privy tower, bedrobe hiked up around his hips. […]
For once, his father did what Tyrion asked him. The proof was the sudden stench, as his bowels loosened in the moment of death. Well, he was in the right place for it, Tyrion thought. But the stink that filled the privy gave ample evidence that the oft-repeated jape about his father was just another lie. Lord Tywin Lannister did not, in the end, shit gold.
—from Tyrion’s assassination of Tywin during his escape from the black cells
The King’s Hand was rotting visibly. His face had taken on a greenish tinge, and his eyes were deeply sunken, two black pits. Fissures had opened in his cheeks, and a foul white fluid was seeping through the joints of his splendid gold-and-crimson armor to pool beneath his body. […]
Red-eyed and pale, Cersei climbed the steps to kneel above their father, drawing Tommen down beside her. The boy recoiled at the sight, but his mother seized his wrist before he could pull away.“Pray,” she whispered, and Tommen tried. But he was only eight and Lord Tywin was a horror. One desperate breath of air, then the king began to sob.“Stop that!” Cersei said. Tommen turned his head and doubled over, retching. His crown fell off and rolled across the marble floor. His mother pulled back in disgust, and all at once the king was running for the doors, as fast as his eight-year-old legs could carry him. “Ser Osmund, relieve me,” Jaime said sharply, as Kettleblack turned to chase the crown. He handed the man the golden sword and went after his king. In the Hall of Lamps he caught him, beneath the eyes of two dozen startled septas. “I’m sorry,” Tommen wept. “I will do better on the morrow. Mother says a king must show the way, but the smell made me sick.” This will not do. Too many eager ears and watching eyes.“Best we go outside, Your Grace.” Jaime led the boy out to where the air was as fresh and clean as King’s Landing ever got. Twoscore gold cloaks had been posted around the plaza to guard the horses and the litters. He took the king off to the side, well away from everyone, and sat him down upon the marble steps. “I wasn’t scared,” the boy insisted. “The smell made me sick. Didn’t it make you sick? How could you bear it, Uncle, ser?”
—Jaime, Cersei, and Tommen, during Tywin’s funeral.
I’m just saying: means, motive, opportunity.
The means being the expansive knowledge of poisons.
The motive being the order to kill Elia and her Children during the sack of King's Landing, which Oberyn is absolutely convinced was given by Tywin.
The opportunity being breakfast with Tywin and Mace Tyrell.
No, he did not poison him (as far as we know).
Nothing like that was mentioned neither on the books or the show. We never saw him trying to do something like that.
He might have been, but we see Tywin being killed by his son, and not by poisoning, so we will never know unless something is revealed later on.
Elio and Linda, the co-authors of The World of Ice and Fire, have said no. Although their word is not canon (even George's word isn't) I believe they know enough of the subject and have enough authority in the series to be worth listening too.
No, Tywin Lannister was not poisoned.
/r/asoiaf, AMA with authors of The World of Ice and Fire, Elio García and Linda Antonsson! (Part 1 of 2)