Labeling objects with pixel-wise segmentation requires a huge amount of human labor compared to bounding boxes. Most existing methods for weakly supervised instance segmentation focus on designing heuristic losses with priors from bounding boxes. While, we find that box-supervised methods can produce some fine segmentation masks and we wonder whether the detectors could learn from these fine masks while ignoring low-quality masks. To answer this question, we present BoxTeacher, an efficient and end-to-end training framework for high-performance weakly supervised instance segmentation, which leverages a sophisticated teacher to generate high-quality masks as pseudo labels. Considering the massive noisy masks hurt the training, we present a mask-aware confidence score to estimate the quality of pseudo masks and propose the noise-aware pixel loss and noise-reduced affinity loss to adaptively optimize the student with pseudo masks. Extensive experiments can demonstrate the effectiveness of the proposed BoxTeacher. Without bells and whistles, BoxTeacher remarkably achieves 35.0 mask AP and 36.5 mask AP with ResNet-50 and ResNet-101 respectively on the challenging COCO dataset, which outperforms the previous state-of-the-art methods by a significant margin and bridges the gap between box-supervised and mask-supervised methods. The code and models will be available later.