Oddly, we know at least as much about how Polaris will eventually die as we know about how it was born, despite the fact that it hasn't died yet. This is, at least in part, because the theory behind star death is somewhat clear-cut and simple, whereas we really don't know that much about how to predict the birth of stars.
As predicted in the last section of this document, Polaris will run out of fuel in about 5 million years, give or take a billion or so. Polaris just passes the threshold to be named a 'Massive Star'; this threshold is 3 solar masses, and Polaris is estimated to be 4. When Polaris runs out of fuel, it will become a red supergiant:
A red giant. Or just the sun through a filter; I don't really know if I trust the source for this picture.
It will blow off its outer shell, and its core will start burning helium, and then heavier elements, until it is primarily iron. AFter about a million years, the core will, more or less, spontaneously implode, creating a supernova.
This supernova will be very dramatic; the North Star will likely take up a good portion of the sky. Eventually, Polaris's remaining core will contract, and most likely become a neutron star. The cutoff for a neutron star is just over 3 solar masses, and Polaris will likely blow off at least one solar mass with its shell when it goes supernova. If it does not, it will become a black hole; however, in this case, I find this quite unlikely; in all probability, Polaris will end up as a superdense chunk of what was once, before it became one great big atomic nucleus, iron.