DNA denaturation has long been a subject of intense study due to its relationship to DNA transcription and its fundamental importance as a structural phase transition. Many aspects of this phenomenon, however, remain poorly understood. Existing models fit quite well with experimental results on the fraction of unbound base pairs versus temperature. Yet, these same models give incorrect results for other essential quantities. For example, the predicted base pair fluctuation timescales - relevant to transcription - are orders of magnitude different from experimental ones. Here, we demonstrate that nanoscale thermal transport can serve as a sensitive probe of the underlying microscopic mechanisms responsible for the dynamics of DNA denaturation. Specifically, we show that the heat transport properties of DNA are altered significantly and abruptly as it denaturates, and this alteration encodes detailed information on the dynamics of thermal fluctuations and their interaction along the chain. This finding allows for the unambiguous discrimination between models of DNA denaturation. Measuring the thermal conductance will thus shed new light on the nonlinear physics of this important molecule.