In near-field thermophotovoltaics, a substantial enhancement of the electrical power output is expected as a result of the larger photogeneration of electron-hole pairs due to the tunneling of evanescent modes from the thermal radiator to the photovoltaic cell. The common low-injection approximation, which considers that the local carrier density due to photogeneration is moderate in comparison to that due to doping, needs therefore to be assessed. By solving the full drift-diffusion equations, the existence of high-injection effects is studied in the case of a GaSb p-on-n junction cell and a radiator supporting surface polaritons. Depending on doping densities and surface recombination velocity, results reveal that high-injection phenomena can already take place in the far field and become very significant in the near field. Impacts of high injection on maximum electrical power, short-circuit current, open-circuit voltage, recombination rates, and variations of the difference between quasi-Fermi levels are analyzed in detail. By showing that an optimum acceptor doping density can be estimated, this work suggests that a detailed and accurate modeling of the electrical transport is also key for the design of near-field thermophotovoltaic devices.