At the end of "The Birds," Nat throws his empty cigarette packet onto the fire and watches it burn. This moment is indeed symbolic: that the packet is empty, for example, signifies that humankind is all out of ideas in their battle against the birds. Mr Trigg, for instance, tried to fend them off with this gun but lost his life in the process. Planes deployed by the government did not fare any better, either.
The empty packet also symbolises the end of society as Nat knows it: Mr and Mrs Trigg are dead, his family is in hiding from flocks of violent birds and the BBC's radio programmes have disappeared. By watching the packet burn in the fire, Nat is accepting these social changes and looking forward to the future, however uncertain it may be. The burning packet is also symbolic of Nature's victory over humans: through their repeated attacks, the birds have brought humans to their knees. The burning packet is thus a potent symbol of this violence and destruction.