That depends on the fish doing the eating, Perciform fish such as Cichlids and bass, siamese tigers, leaf fish etc have a set of teeth at the back of the throat known as the pharnegal mill, anything that passes through this is mashed to a pulp so the food is well and truely dead before it reaches the stomach, this trait is also shared by the Cyprinid family though these tend not to eat fish live.
Catfish on the other hand tend just to gulp and swallow so it is quite likely that the prey remains alive for a short while before the digestive juices set to work.
