Thermal cameras measure the temperature of objects based on radiation emitted in the infrared spectrum. In this work, we propose a novel shape recovery approach that exploits the properties of heat transport, specifically heat conduction, induced on objects when illuminated using simple light bulbs. While the resulting heat transport occurs in the entirety of an object's volume, we show a surface approximation that enables shape recovery and empirically analyze its validity for objects with varying thicknesses. We develop an algorithm that solves a linear system of equations to estimate the intrinsic shape Laplacian for the first time from thermal videos along with several properties including heat capacity, convection coefficient, and absorbed heat flux under uncalibrated lighting of arbitrary shapes. Further, we propose a novel shape from Laplacian objective that aims to resolve the inherent shape ambiguities by drawing insights from absorbed heat flux images using two unknown lights sources. Finally, we devise a coarse-to-fine refinement strategy that faithfully recovers low- and high-frequency shape details. We validate our method by showing accurate reconstructions, to within an error of 1-2 mm, in both simulations and from noisy thermal videos of real-world objects with complex shapes and materials.