We propose a solid state heat pump based on photon assisted heat transfer between two large-area light emitting diodes coupled by the electromagnetic field and enclosed in a semiconductor structure with a nearly homogeneous refractive index. Ideally the thermophotonic heat pump (THP) allows heat transfer and electricity generation at the Carnot efficiency, but in reality there are several factors that limit the efficiency. We present a numerical model that accounts for the most important losses of the thermophotonic heat pump to study the operating regimes and the fundamental limitations of the THP structure. The results show that the thermophotonic heat pump has potential to outperform heat pumps based on the thermoelectric effect especially for heat transfer across large temperature differences. In energy harvesting applications the performance of the THP is good for small power densities, but drops at high power densities unless the losses in the structure can be efficiently minimized.