Plasmonic absorption of light can lead to significant local heating in metallic nanostructures, an effect that defines the sub-field of thermoplasmonics and has been leveraged in diverse applications from biomedical technology to optoelectronics. Quantitatively characterizing the resulting local temperature increase can be very challenging in isolated nanostructures. By measuring the optically-induced change in resistance of metal nanowires with a transverse plasmon mode, we quantitatively determine the temperature increase in single nanostructures, with the dependence on incident polarization clearly revealing the plasmonic heating mechanism. Computational modeling explains the resonant and nonresonant contributions to the optical heating and the dominant pathways for thermal transport. These results, obtained by combining electronic and optical measurements, place a bound on the role of optical heating in prior experiments, and suggest design guidelines for engineered structures meant to leverage such effects.