which animal will evolve to play chess with humans? monkeys or something else?

No animals, because we've already (according to Stanford) entered the "sixth great mass extinction" event.

Just throwing in an alternate point of view:
http://www.bbc.com/earth/story/20150818-chimps-living-in-the-stone-age
They are making tools on their own as the need arises. Which means in a couple of years they will beat me in chess.

In the world of variant chess we already have bulldogs and zombies on the chessboard.
But zombies aren't really an animal - they're actually undead people.

A pet playing chess against its owner is a cool sci-fi concept. I'm a writer, so I see great appeal in that aspect of this inquiry. As a writer, I'd love to imagine a scene like that in my next excursion to the science fiction realm with my novellas. But that's a little off topic for the question, so I may as well give my opinion.
I think the potential of studying apes' intelligence is unappreciated by the scientific community. We share 99% of our D.N.A. with chimpanzees, and at least 90% of it with most apes. In my opinion, studying them, especially their intelligence, would lead to great developments in our knowledge of humans, human intelligence, and perhaps even some aspects of the human condition. If we share 99% of our D.N.A. with chimpanzees, there's a 99% chance we could teach them chess with time. Who knows? Maybe one day, the World Chess Champion won't be human.
Edit: That was kinda wordy

Pigeons have been taught to target underwater mines...
So who knows... maybe teach them the London System.
Octopus , dolphin or wait... maybe a Whale (don't get too close when they morse code the movement of their pieces! )
Maybe in a semi-scifi way : a computer-connected cybernetic trilobyte if we refer to the longest reigning species in an era of Earth history! Imagine!
But like a honey badger for most animal buff answers, I'll go with :
A wolverine! It would be a scary opponent, not cause it could play the Najdorf Sicilian, but cuz it would do what a wolverine does and slaugther anything that's under a grizzly bear in size with ease. They'd kidnap your pieces, as they did autochton offsprings as the tales of the Elders say.
As far as I know there are already some Chelsea Football fans who play chess.