Could electrons tunnel into protons in a white dwarf and turn it into neutron star? Just like the Sun, where hydrogen ions tunnel into each other to start fusion, which would be otherwise impossible with this level of kinetic energy, could electrons inside a white dwarf tunnel into protons and turn it into a neutron star? Actually I want to know if the Pauli Exclusion Principle can prevent quantum tunneling so I used a compact star as an example.
 A: The process you describe is called inverse beta decay or neutronisation. No tunnelling is required because there is no potential barrier keeping protons and electrons apart. The Pauli Exclusion Principle isn't relevant because protons and electrons are distinguishable particles.
There is however an energy deficit; the rest mass of a neutron is greater than the combined masses of an electron and a proton. The electrons therefore need to be mildly relativistic to make the reaction go ahead.
However, there are no free protons in a white dwarf interior, they are instead inside carbon and oxygen nuclei. The energy penalty to convert a proton to a neutron in these stable nuclei is even higher and the electrons need to be highly relativistic (e.g. an electron Lorentz factor $\gamma \sim 27$ for neutronisation of carbon nuclei).
Such conditions occur at very high densities in the degenerate electron gas in a massive white dwarf. When the electron Fermi energy reaches the threshold energy for neutronisation, then free electrons are removed from the gas, softening the equation of state and either leading to the collapse of the white dwarf (into a neutron star) or its thermonuclear ignition and destruction in a type Ia supernova.
