Biological structures rely on kinetically tuned charge transfer reactions for energy conversion, biocatalysis, and signaling as well as for oxidative damage repair. Unlike man-made electrical circuitry, which uses metals and semiconductors to direct current flow, charge transfer in living systems proceeds via biomolecules that are nominally insulating. Long-distance charge transport, which is observed routinely in nucleic acids, peptides, and proteins, is believed to arise from a sequence of thermally activated hopping steps. However, a growing number of experiments find limited temperature dependence for electron transfer over tens of nanometers. To account for these observations, we propose a temperature-independent mechanism based on the electric potential difference that builds up along the molecule as a precursor of electron transfer. Specifically, the voltage changes the nature of the electronic states away from being sharply localized so that efficient resonant tunneling across long distances becomes possible without thermal assistance. This mechanism is general and is expected to be operative in molecules where the electronic states densely fill a wide energy window (on the scale of electronvolts) above or below the gap between the highest-occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). We show that this effect can explain the temperature-independent charge transport through DNA and the strongly voltage-dependent currents that are measured through organic semiconductors and peptides.