Strictly speaking no. They can't even pass a written or driving exam. In other words, they can't be issued driving license. Hence, they can't drive.
Some species of monkeys are very intelligent and if you teach them to drive a car, they can easily could. You might have seen them driving in some movies. But one thing is sure that they are not expert drivers. They can only drive to make you happy.
If monkey also can drive,then how about other animals?if the monkey do really have knowledge,why we don't invite monkey to have class for study??
If a monkey can cook and serve dinner on table than why he can't drive a car. The most intelligent in species of monkeys is chapansee which could easily be trained and is slightly powerful than other species. Whatever you can bring accuracy in his driving cause after all its an animal.