How do neutron stars burn? Is it decay or fusion or something else?

With normal stars, the "burning", that is to say the fusion reaction, produces a pressure that counteracts the pull of gravity to keep the star from collapsing. But with neutron stars, the protons and electrons in the star have combined into neutrons*.

The Pauli exclusion principle causes the neutrons to resist further compression. That is, the neutrons, being identical fermions, can't all be put in the same state. So to get them closer and closer together you have to go into higher and higher energy states. Thus, there is an energy cost in compressing the star, and this results in a sort of pressure called "degeneracy pressure".

It is this pressure that stabilizes the neutron star against collapse (assuming it doesn't have enough mass to overcome this pressure and become a black hole). So they don't need to "burn" to maintain their stability, and so far as I know, they don't. At least not in the sense of a normal star where you have atomic nuclei fusing.

  • Note: Neutrons aren't made of protons and elections, but this transformation can happen by means of the weak nuclear force. Normally neutrons aren't stable outside of the atomic nucleus -- instead the transformation would go the other way and a free neutron would decay into a proton and electron (there's also an anti-electron neutrino produced). But under the intense gravitational pressure in a collapsed star, the neutrons are stable, which allows us to end up with neutron stars.

Edit: This is of course a very approximate picture. The link posted by Thomas Thernel has much more detail. One good point to emphasize is that, as you might expect, the density is greater at the center of a neutron star than at its outskirts, so the star won't really be all neutrons... you'll have more neutrons closer to the center, and more ordinary atomic nuclei further out. Apparently some interesting sorts of structures can form from the remaining nuclei, even at the point where it's 90-95% neutrons.

Tim Goodman's answer is right, and I just want to add a couple of things:

Though Neutron stars do not "burn" anything, for the most part, they DO shine, thanks to the fact that they are the remnants of the core of a star, and thus, when they are born, are roughly as hot as the center of a star. They will thus shine in the same way that a hot poker shines when you stick it in a fire.

Second, neutron stars have a maximum possible mass, dictated by the rules of general relativity and the mass of a neutron. If a neutron star has extra mass beyond this dumped on its surface (say, because it is orbiting an ordinary star, and that star accretes mass onto the NS), they can explode in a violent way. To my understanding, this has not been observed, but it has been known to happen in White Dwarfs, which are held together by a very similar principle to neutron stars. The end result is a supernova, and then a black hole.

The correct comparison for a neutron star is with a cinder spat out of the fire. The cinder will glow brightly for a short period of time and then fade rapidly. Such is the fate of neutron stars, because although born at $10^{11}$ K in the heart of a supernova, they have an extremely low heat capacity.

Contrary to common belief - neutron stars are not supported by neutron degeneracy pressure. Yes, this contributes, but a star supported solely by NDP would be limited to masses less than 0.7$M_{\odot}$. All known neutron stars are more massive than this. It is the strong nuclear force in incompressible, asymmetric nuclear matter that provides most of the support.

Neutron degeneracy is though largely responsible for the thermal properties of neutrons stars. Degenerate matter has an extremely low heat capacity, because there are no empty energy states for fermions to cool and fall into. That means that even though they are born with extremely high temperatures, they will be nearly isothermal (degenerate fermions also have long mean free paths and high thermal conductivity), with very little thermal energy.

Neutrino processes (URCA process, followed by modified URCA process - see What allows the modified Urca process to work at lower density than direct Urca in neutron star cooling?) can effectively and rapidly cool the star on timescales of initially seconds and later tens or hundreds of years. After about ten thousand years cooling from the surface by photons dominates.

As the neutron star cools, what distinguishes it from a normal star, is that its internal pressure is independent of temperature. That means that if it is in equlibrium once it cools to say a billion degrees (maybe after a year), it is already "effectively cold" and further cooling will not change its radius and density.

If we assume that most supernvovae from stars with initial masses between 8 and 20 solar masses produce neutron stars then there are probably about a billion of these dead cinders floating around in our Galaxy.