Beryllium polynitride (BeN4) has been recently synthesized under high-pressure conditions [Bykov et al. Phys. Rev. Lett. 2021, 126, 175501]. Its anisotropic lattice structure dependent on the applied pressure motivates exploration of its thermal transport properties with a theoretical framework that combines the Boltzmann transport equation with ab initio calculations. The bonding anisotropy (impacting the phonon and electron group velocities) and bonding anharmonicity (captured through three- and four-phonon scatterings) are reflected in the strong anisotropy of both phononic and electronic components of the thermal conductivity. Moreover, the pressure-driven evolution of the interlayer Be-N bonding, from partially covalent (under high-pressure synthesis conditions) to van der Waals (under ambient pressure), drives a largely interlayer thermal conductivity. These findings highlight an alternative strategy for achieving directional control of the thermal transport in synthetic materials.