Three-dimensional numerical calculations based on the finite element method are performed to calculate the increase in the temperature in nanostructured cells of a magnetic tunnel junction under conditions that are relevant to current-induced magnetization switching for a high-density magnetic random access memory. Three key parameters, the lateral size, the resistance-area product and the applied current density, were varied widely so that their effects on the temperature increase could be examined. The computed results for the temperature increase, as a function of the resistance-area product and the current density, show the same trends that are expected from an equation for the dissipated heat. While the increase in the temperature is expected to be independent of the lateral size, the computations reveal a rather complicated relationship between the two variables, which is contingent on the various conditions that are considered. In a cell array that is relevant to high-density contexts, the temperature increase in the nearest cells is as high as 50% of the cell at which the current is directly applied; this could cause a thermal-stability problem in high-density magnetic random access memories. The temperature increase was also calculated under a more realistic physical picture of the relaxation of tunnelled electrons. These results are in agreement with those that are computed from Joule heating.