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 inelastic-dissipative 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.