We study the transport mechanisms of different types of azurin (Az) monolayer heterojunctions with a variety of metal substituents. The systems include Holo-Az (Cu-substituted), Apo-Az (no metal), and Ni-, Co- and Zn-substituted azurins. Our theoretical analysis is based on measurements of the voltage and temperature dependencies of the current and attempts to reproduce both dependencies using a common mechanism and corresponding set of parameters. Our results strongly suggest that for Holo-Az the transport mechanism depends on the protein monolayer/heterojunction setup. In one type of heterojunction, transport is dominated by resonant incoherent hopping through the Cu redox site, whereas in the other it is mediated by off-resonant tunneling. For the unsubstituted (Apo-Az) and other metal-substituted azurins, the dominant transport mechanism at low temperatures is off-resonant tunneling, with an average tunneling barrier that depends on the type of metal dopant, and at the highest temperatures, it is through-amino-acid hopping. Our modeling results are relevant to the analysis of the current behavior over a range of temperatures for any molecular heterojunction device.