Pulsars (and neutron stars in general) are already “dead” stars. They are the end states of stars that start out with 10 to 30 solar masses. They do not fuse elements in their core.
Left on its own, a pulsar will go through complicated phases of cooling. It will also spin down over time. However, nothing more interesting will happen.
However, given the right conditions a pulsar can collapse further into a black hole. Pulsars are held up against gravitational collapse by neutron degeneracy pressure (How does the Pauli Exclusion Principle relate to degeneracy pressure?). A pulsar can overcome this and collapse further if it acquires a red giant partner from which it can accrete mass. If it accretes enough mass to break the Tolman-Oppenheimer-Volkov limit of about three solar masses, it will collapse into a black hole.