First some context:
A cold iron "white dwarf" will be stable if it's mass is below about $1.2M_{\odot}$. This is the equivalent of the Chandrasekhar mass for a more normal white dwarf supported by electron degeneracy pressure, but is lower because there are fewer electrons per mass unit in a gas of iron.
However, lower energy configurations are possible - i.e. neutron stars and black holes if the electron degeneracy can be compromised.
One way of doing this is electron capture (a.k.a. neutronisation or inverse beta decay). This is where a proton in the iron nucleus captures an electron from the degenerate gas and turns into a neutron.
The process is "endothermic", the electron requires an energy of around 5 MeV. This can be achieved in a degenerate electron gas if the electron density is high enough to push the Fermi energy beyond this threshold.
If you do the sums, these densities are also reached just a little below $1.2 M_{\odot}$ (recall that more massive white dwarfs get smaller).
This process occurs in the crusts of neutron stars and builds up increasingly neutron-rich nuclei (note that iron is only the equilibrium nucleus in low density matter). However, it is not possible to build an entire star out of material behaving in this way because the removal of free electrons softens the equation of state dramatically and leads to collapse.
Now the answer:
In an iron white dwarf with mass below $1.2 M_{\odot}$ we would usually assume stability because the electron Fermi energy is below the neutronisation threshold. However, quantum tunneling will mean that occasionally, an electron capture will occur, leading to a more neutron-rich nucleus and one fewer electron in the gas. This lowers the degeneracy pressure, the star gets smaller to compensate and the electron density increases again. Repeat many times and eventually the material attains too few electrons per unit mass to support itself and it collapses.
In other words the tunneling gradually makes more neutron-rich material and lowers the Chandrasekhar mass until instability is triggered.
The result would be a neutron star.
I don't understand at all what the process to form black holes could be. Any long-lived, cold iron star must be less massive than $1.2 M_{\odot}$ as explained above. The collapse of such an object will always lead to a neutron star because this is well below the maximum mass that can be supported by dense neutron star matter.
Dyson's arguments in the paper referred to are very abstract (and unconvincing to me - for example he says there is no stable state for stars greater than the Chandrasekhar mass, although the first neutron star mass measurements, showing that they could exist with greater masses were not available in 1978). It may be possible to randomly form tiny black holes that then end up consuming the entire star. On the other hand, it is widely thought that tiny black holes will immediately evaporate in a burst of radiation. So this process seems unlikely to turn the whole star into a black hole and instead just gradually convert the rest mass into Hawking radiation.
Dyson does not directly mention quantum tunneling as anything that facilitates this process, though given that nucleons are repulsive at short range it would likely be required. The range of suggested timescales in Dyson's paper are connected with how small the smallest black hole can be - whether it is the Planck mass or the somewhat higher minimum mass for which a classical black hole description is meaningful and hence on the number of nucleons that have to be crammed into their Schwarzschild radius in order for this to happen.
