(New edit regarding season 4, episode 3 of the TV-show at the bottom)
It's not the first time Tywin kills a mad king (well, he meant to when he sacked King's Landing). That alone should tell you something. Moreover, lets look at the signs:
Tywin recognizes that Cersei is allowing Joffrey to make huge blunders in King's Landing, such as killing Ned Stark, firing Ser Barristan, etc, and he sends Tyrion to take charge of things.
In a specific confrontation after the battle of the Blackwater, Tywin compares Joffrey to Aerys:
“Aerys also felt the need to remind men that he was king. And he was passing fond of ripping tongues out as well. You could ask Ser Ilyn Payne about that, though you’ll get no reply.”
This is after Joffrey wants to serve Sansa Robb's head at his wedding feast. Later he makes a comment about Joffrey being too much Robert and too little Lannister:
"And what were you telling him, pray? I did not fight a war to seat
Robert the Second on the Iron Throne. You gave me to understand the
boy cared nothing for his father."
Lastly, he says ominously that Joffrey requires a "sharp lesson".
"The boy is thirteen. There is time yet." Lord Tywin paced to the
window. That was unlike him; he was more upset than he wished to show.
"He requires a sharp lesson."
Tyrion had gotten his own sharp lesson at thirteen. He felt almost
sorry for his nephew. On the other hand, no one deserved it more.
I think at this point that Tywin realizes that Joffrey is very likely to bring shame on house Lannister and jeopardize the throne. If he made a plot with Walder Frey to assassinate Robb at his own wedding, it makes some kind of poetic sense that he would conspire with the Tyrells to kill Joffrey at his. Not only would it rid him of Joffrey, he could also pick a target to name as his killer, and as Prince Oberyn points out to Tyrion:
“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?”
Oberyn would have a motive as well: Killing Joffrey would not simply be a petty murder, but it would allow for Dorne to crown Myrcella, as she would be Joffrey's heir according to Dornish law.
It was clear that Prince Oberyn wanted Tywin's head on a platter. He knew Tywin was responsible for murdering his sister Elia and her children. It would be a very neat plan indeed to frame Prince Oberyn for Joffrey's murder. If only Cersei had not got the jump on Tywin and accused Tyrion instead. Tywin might have called that only a minor setback, since he never liked Tyrion much.
Also, as a little treat at this point, (Spoiler ASOS, Season 4:3+)
there is the theory that Oberyn had poisoned Tywin, and that he was dying of that poison when Tyrion killed him.
The Tyrells were also not very friendly with Oberyn, and it would have been an easy thing to convince them of this plot. When it turned out to be Tyrion instead... oh well. Tywin would not kill his son, just make him take the black.
What evidence is there to the contrary? That Tywin loves his family? Hardly, he loves his family's reputation, that is all. That he would not plot dishonourable assassinations? Ask Robb Stark about that one. That he would not harm his kin? Ask Tyrion about that.
Would Tywin risk the wrath of Dorne by accusing Oberyn? That's a weak spot in this theory. Prince Doran still holds Myrcella, and they have nothing but the betrothal between her and Trystane Martell to hold firm the peace between them.
Is it more likely that Littlefinger acquired a magic necklace to poison Joffrey? That old Olenna Tyrell tweaked Sansa's hairnet to target Joffrey? We all know that Littlefinger is a liar.
Anyway, it is a neat theory that ties up some of the loose ends, and it is always fun to speculate on. It sounds all but certain that Tyrells were involved, judging by what Olenna Tyrell told Sansa, that the marriage would happen, despite Joffrey being a monster. But would they act alone? Would they trust Littlefinger to be part of such a plot and go behind Tywin's back?
After watching Season 4, Episode 3, the event described in this question, Tywin's history lesson with Tommen, I think demonstrates that it is not at all unlikely that Tywin was involved in Joffrey's murder. He clearly relishes this moment to tutor Tommen, to the point that he does it beside Joffrey's dead body. He even says that Joffrey was not wise, which was the whole point of his history lesson, but Tommen "will be". He will listen to his advisors, as Joffrey would not have, and continue to do so after he is a King in his own right.
It is fairly clear that Tywin was convinced that Joffrey would not have allowed Tywin much control once he was the King.
I believe that Tyrion even mentions it in the scene afterwards.
In this article GRRM himself says that he thinks the murderers (yes, plural "s") thought that Joffrey's death would be seen as an accident, which is consistent with the method of choice that Tywin would have chosen, if he wanted to avoid the scandal of a Lannister being assassinated.