We study the heat kernel and the Green's function on the infinite supercritical percolation cluster in dimension $d \geq 2$ 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. The proof relies on a structure of renormalization for the infinite percolation cluster introduced by Armstrong and the first author, Gaussian bounds on the heat kernel established by Barlow 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.