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.