A holistic way to think about this is to consider the energy density of the gas that makes up a system. In the core of a star like the Sun we could write this down as
$$u = u_p + u_e + u_{\rm He} + u_z,$$
where $u_p$ is the energy density of the protons (including their rest mass), $u_{e}$ ditto for the electrons, $u_{\rm He}$ ditto for the helium nuclei and $u_z$ for the energy density of heavier nuclei. In the Sun, all these particles are non-relativistic, so each terms is the sum of the kinetic energy density of the particles (roughly $nkT$, where $n$ is the number density of those particles) and their rest-mass energy densities (roughly $nMc^2$, where $M$ is the rest-mass of the particles).
To work out what the equilibrium composition is then you would minimise the energy density with respect to the relative compositions, subject to the various conservation laws (charge, baryon number etc.) In turn, the result of this calculation will depend on the density of the gas and its temperature.
In the solar core it turns out the current composition is very far from the equilibrium composition that would minimise the energy density. A much lower energy density can be achieved by turning the hydrogen into helium and then in fact into heavier elements all the way up to iron. Whether an equilibrium can be reached on an interesting timescale then depends on the nuclear physics of the situation - it turns out that for the Sun, the equilibrium composition cannot be reached and even turning hydrogen into helium takes 10 billion years. Nevertheless, the possibility is there and is dramatically realised when a massive C/O white dwarf (the Sun will become a C/O white dwarf) turns most of its nuclei into iron-peak elements in a type Ia supernova explosion.
In neutron stars, the bulk of the object is a fluid containing neutrons, electrons and protons.
$$ u = u_n + u_p + u_e$$
At densities of $\sim 10^{17}$ kg/m$^3$, the equilibrium composition is such that there are mostly neutrons with about a 1% contribution from protons and electrons (that is why they are called neutron stars).
So the key question is whether there is an alternative composition (perhaps at higher densities) that this can turn into on a slow(ish) timescale which has a lower energy density.
The answer to the first part is yes; the answer to the second part is sort-of, but not comparable with the solar nuclear timescale.
There are various types of exotic matter that might be created at higher densities. These include the production of muons, the production of hyperons or a change to a quark phase if quarks have asymptotic freedom at very high densities. All of these are likely to take place very quickly (even the muon production which is moderated by the weak force). Thus the neutron star is essentially born that way. Once the neutrons become completely degenerate then neutron star cooling does not lead to a change in pressure or density, so the composition would essentially be fixed: almost.
There are however composition changes that take place that do result in a more continuous output of energy. The early lives of neutron stars the first (100,000 years or so) are dominated by neutrino cooling. These arise from weak interactions turning neutrons into protons and vice versa. The energy driving these net endothermic reactions comes from the residual heat in the almost degenerate gases as the neutron star settles to its equilibrium composition. There is significant escape of energy in the form of roughly thermal neutrinos and the luminosities can be quite considerable. For instance the modified URCA process gives neutrino luminosities of around a million solar luminosities for a young neutron star, declining to about a solar luminosity after 100,000 years.
The energy density can also be lowered by the formation of Cooper pairs in the degenerate neutron and proton gases. Superfluid pairing occurs at a critical temperature that depends on the density. Since the interior of a neutron star is roughly isothermal, but has a range of densities, this phase change occurs at different times at different radii within the neutron star and releases about 1 MeV per pair of energy in the form of neutrinos. It is likely that this phenomenon too is fairly brief, perhaps only occurring within the first 10,000 years or so of the life of a neutron star.
There are however, no fusion reactions. Sticking baryons together at this point will not lower the energy density of the gas.
