The T-virus from the movie Resident Evil destroys pretty much all of the brain except for the cerebellum and brain stem. Basically this reverts a person back to the most primal state with poor motor skills. I guess when you break a person down, the only thing they really know is that they have to eat to survive. I suppose the point I'm trying to make is that you can't really learn anything without a brain. However if you can remember the instinct to feed then is there a possibility that you can remember other things.
In the 1985 film, Day of the Dead, Dr Logan believe that zombies can become domesticated and achieves this with a zombie he calls Bub. Bub seems to remember parts of its former life and certain objects. One of the soldiers becomes outraged and kills one of the doctors, Bub sees this and appears to become upset and even sheds a tear.
They touch base on this again in The Land of the Dead, 2005. The zombies in town seem to remember their former lives. They can be seen playing instruments and operating gas pumps, although not very well. A particular zombie named "Big Daddy" is seen getting upset at the sight of his fellow zombies getting killed (or rekilled I guess you could say) and eventually learns to use firearms. He teaches his fellow undead to use weapons and eventually the go for revenge.