What happens to the nerve impulse after the person dies?

Nothing "happens" to it. The electrical energy of nerve impulses is generated by chemical processes in a living body. When the body dies the energy is no longer available. Like if you destroy a battery or let it run down.

Contrex is right. It's basically dissipated as heat.

The electric impulse is generated via chemical energy which is mediated via ion exchange mechanisms across the nerve membranes. Once, the person dies, the pumps are no longer maintained, regulated ion flux decreases and impulses can no longer be generated or conducted.

The net result is a conversion to a state of entropy, manifested by an increase in heat production and loss of membrane integrity.

....either that ....or ectoplasm in the back of the net at the Stretford End !