This Letter presents an analysis of three-dimensional vector currents and temperatures observed in a sunspot from the photosphere to the chromosphere, spanning a range of heights of approximately 1500 km. With this unique data set, based on novel spectropolarimetric observations of the 850 nm spectral region, it is possible to conduct for the first time an empirical study of the relation between currents and chromospheric heating. It is shown that while resistive current dissipation contributes to heat the sunspot chromosphere, it is not the dominant factor. The heating effect of current dissipation is more important in the penumbra of the sunspot, but even there it is still a relatively modest contribution.