We study the heat kernel and the Green's function on the infinite supercritical percolation cluster in dimension d ≥ 2 and prove a quantitative homogenization theorem for these functions with an almost optimal rate of convergence. These results are a quantitative version of the local central limit theorem proved by Barlow and Hambly in [23]. The proof relies on a structure of renormalization for the infinite percolation cluster introduced in [12], Gaussian bounds on the heat kernel established by Barlow in [21] and tools of the theory of quantitative stochastic homogenization. An important step in the proof is to establish a C 0,1 -large-scale regularity theory for caloric functions on the infinite cluster and is of independent interest.