We present a new method for nanoscale thermal imaging of insulating thin films using atomic force microscopy (AFM). By sweeping the voltage applied to a conducting AFM tip in contact mode, we measure the local current through a VO$_2$ film. We fit the resultant current-voltage curves to a Poole-Frenkel conduction model to extract the local temperature of the film using fundamental constants and known film properties. As the local voltage is further increased, the nanoscale region of VO$_2$ undergoes an insulator-to-metal transition. Immediately preceding the transition, we find the average electric field to be 32 MV/m, and the average local temperature to be at least 335 K, close to the bulk transition temperature of 341 K, indicating that Joule heating contributes to the transition. Our thermometry technique enables local temperature measurement of any film dominated by the Poole-Frenkel conduction mechanism, and provides the opportunity to extend our technique to materials that display other conduction mechanisms.