What happens over time as a neutron star cools? My understanding of a white dwarf star is that, although it hasn't happened yet due to the age of the universe, eventually as it loses energy due to radiating heat and tidal forces, it will cool to become a black dwarf - essentially a black ball of carbon.

What happens when a neutron star eventually cools? What is left, and do we know what it would look like?
The cooling history of a neutron star can be divided into an extremely rapid neutrino cooling phase, followed by an indefinitely long cooling phase due to the emission of photons from its surface.

The key to understanding neutron star cooling is to realise that degenerate neutrons possess almost no thermal energy, even when extremely hot - because the Pauli Exclusion principle forbids the neutrons from losing kinetic energy and falling into an already-occupied quantum state. Any cooling processes therefore act to reduce the temperature of the neutron star very quickly. Secondly, the interior of a neutron star is almost isothermal, due to the high thermal conductivity of degenerate gases, but the temperature at the surface is smaller by about a factor of 100 or so.

When a neutron star first forms, its internal temperature exceeds 10 billion kelvin, its surface temperature would be 100 million degrees and emit hard X-rays. Nevertheless, the small size of the neutron star limits radiative losses and instead it is neutrinos escaping from the interior that can cool it by a factor of around 100 in a few thousand years.

The neutrino losses proceed by something called the modified URCA process (somewhat faster direct URCA or processes involving pions/kaons maybe important in the first seconds/minutes of more massive neutron stars, but are blocked by complete neutron degeneracy and the comparative paucity of protons in the interior): cycles of neutron beta decay and inverse beta decay on protons produce anti-neutrinos and neutrinos. Bystander particles are required to conserve momentum. The temperature dependence of this process is $\sim T^{8}$, so if the temperature falls by a factor 100, the neutrino cooling rate falls by a factor of $10^{16}$. 

Meanwhile, photon cooling from the surface, whilst initially negligible compared to neutrino cooling, falls only as $T^{4}$. After around 10,000-100,000 years, when the neutron star surface has cooled to about a million degrees, photon cooling, through soft X-ray emission dominates.

If nothing else happened to an isolated neutron star, it would continue to cool, such that $T \propto t^{-1/2}$. The vast majority of the estimated 1 billion neutron stars in our Galaxy, are in this state, whereas most of the pulsars we can observe are young and in the neutrino-cooling phase.

How cold can they get? The formula above suggests their temperature halves when their age quadruples. Following this, they could cool to the temperature of the Sun in around a billion years. However, there are reheating processes that can occur.

Isolated neutron stars can accrete material from the interstellar medium. This may be very little, since most neutron stars move fast. The gravitational potential energy is partly released as heat when it impacts the neutron star surface. Then, the low heat capacity of neutrons works in the opposite direction and comparatively little accretion is required to balance the photon losses.

Secondly, neutron stars are born as fast rotators and spin down. Angular momentum must be transferred outwards from the fluid interior to the outer crust. This is not a frictionless process. Thus some of the rotational kinetic energy can keep the star warm.

Thirdly, the neutron star has a huge magnetic field that gradually dissipates. Some of that energy may end up as heat in the neutron star through simple Ohmic dissipation of currents.

The bottom line is that the thermal emission from neutron stars older than about 100,000 years has not been observed, so we simply don't know how cool they might get. Some more information and references can be found in the Physics SE answer where I discuss where old neutron stars might appear on the HR diagram.