Why can't neutron stars ignite and explode? Beyond the Chandrasekhar limit, white dwarfs become extremely hot. As a result, previously unfusable carbon can become fusable, causing nuclear reactions. This leads to a thermal runaway and ultimately a type Ia supernova.
However, this doesn't seem to happen in neutron stars. They can keep gaining mass until they become black holes. After a certain point, shouldn't the immense temperatures cause a thermal runaway like in white dwarfs, leading to their explosion?
In a white dwarf, the dense matter is not in its lowest energy configuration. Energy can still be extracted from the white dwarf material by fusion, provided it can be ignited.

What exothermic nuclear reactions would there be that could take place in a neutron star? The bulk of the material is in the form of neutrons with a small number of protons and electrons. At these densities, that is the most stable equilibrium composition. 

If a neutron star gains mass in a gradual way, then the most likely course of events will be that its radius will decrease (that is what happens in objects supported by degenerate matter) until it reaches a General Relativistic instability where its collapse to a black hole is inevitable (when $R$ is somewhere between 1.25 and 2 times the Schwarzschild radius). It is possible that neutrons may transform before that into additional hadronic degrees of freedom or into quark matter, but these are endothermic processes that suck kinetic energy out of the neutron gas and only hasten the collapse.