We extend the Landauer-Büttiker probe formalism for conductances to the high bias regime, and study the effects of environmentally-induced elastic and inelastic scattering on charge current in single molecule junctions, focusing on high-bias effects. The probe technique phenomenologically incorporates incoherent elastic and inelastic effects to the fully coherent case, mimicking a rich physical environment at trivial cost. We further identify environmentally-induced mechanisms which generate an asymmetry in the current, manifested as a weak diode behavior. This rectifying behavior, found in two types of molecular junction models, is absent in the coherent-elastic limit, and is only active in the case with incoherent-inelastic scattering. Our work illustrates that in the low bias -linear response regime, the commonly used "dephasing probe" (mimicking only elastic decoherence effects) operates nearly indistinguishably from a "voltage probe" (admitting inelasticdissipative effects). However, these probes realize fundamentally distinct I-V characteristics at high biases, reflecting the central roles of dissipation and inelastic scattering processes on molecular electronic transport far-from-equilibrium.