In this paper, we consider the inverse graph filtering process when the original filter is a polynomial of some graph shift on a simple connected graph. The Chebyshev polynomial approximation of high order has been widely used to approximate the inverse filter. In this paper, we propose an iterative Chebyshev polynomial approximation (ICPA) algorithm to implement the inverse filtering procedure, which is feasible to eliminate the restoration error even using Chebyshev polynomial approximation of lower order. We also provide a detailed convergence analysis for the ICPA algorithm and a distributed implementation of the ICPA algorithm on a spatially distributed network. Numerical results are included to demonstrate the satisfactory performance of the ICPA algorithm in graph signal denoising.