"Dumb beasts" were wild animals that lacked higher-level intelligence, and mostly obviously, the ability to speak or comprehend human language. Except for humans themselves, all animals from Earth could be considered dumb beasts; however, the term was used exclusively in the World of Narnia where such creatures were differentiated from a group of sentient animals known as talking beasts.
At the Narnia's creation, all animals were sentient, though Aslan allowed the animals to choose whether they wished to maintain their superior intelligence. Dumb beasts were those who had "chosen not to speak", or, in effect, decided to lose their sentient capabilities for knowledge. By Aslan's orders, these dumb beasts were then to be looked after by talking beasts and talking beasts to be ruled by humans.
It is unknown if dumb beasts had souls like their talking counterparts. If so, it can be presumed that their souls entered Aslan's Country upon death.