Nonlocal low-rank (LR) tensor modeling has shown great potential in hyperspectral image (HSI) denoising, which first uses the nonlocal self-similarity (NSS) prior to search for many similar full-band patches to form three-dimensional nonlocal full-band groups (tensors), and then usually enforces an LR penalty on each nonlocal full-band group. However, in most existing methods, the LR tensor is only approximated directly from the degraded nonlocal full-band tensor, which is subject to certain issues (e.g., in heavy noise environments) in obtaining a suboptimal tensor approximation, and thus leading to unsatisfactory denoising results. In this paper, we propose a novel nonlocal rank residual (NRR) approach for highly effective HSI denoising, which progressively approximates the underlying L-R tensor via minimizing the rank residual. Towards this end, we first obtain a good estimate of the original nonlocal full-band group by using the NSS prior, and then the rank residual between the de-graded nonlocal full-band group with the corresponding estimated nonlocal full-band group is minimized to achieve a more accurate LR tensor. Moreover, the global spectral LR prior is employed to reduce the spectral redundancy of HSI in the proposed denoising framework. Finally, we develop a simple yet effective alternating minimization algorithm to jointly refine global spectral information and nonlocal full-band groups. Experimental results clearly show that the proposed NRR algorithm outperforms many state-of-the-art HSI denoising methods. The source code of the proposed NRR algorithm for HSI denoising is available at: https://github.com/zhazhiyuan/NRR_HSI_Denoising_Demo.git.