Whole Slide Image (WSI) analysis, with its ability to reveal detailed tissue structures in magnified views, plays a crucial role in cancer diagnosis and prognosis. Due to their giga-sized nature, WSIs require substantial storage and computational resources for processing and training predictive models. With the rapid increase in WSIs used in clinics and hospitals, there is a growing need for a continual learning system that can efficiently process and adapt existing models to new tasks without retraining or fine-tuning on previous tasks. Such a system must balance resource efficiency with high performance. In this study, we introduce COSFormer, a Transformer-based continual learning framework tailored for multi-task WSI analysis. COSFormer is designed to learn sequentially from new tasks wile avoiding the need to revisit full historical datasets. We evaluate COSFormer on a sequence of seven WSI datasets covering seven organs and six WSI-related tasks under both class-incremental and task-incremental settings. The results demonstrate COSFormer's superior generalizability and effectiveness compared to existing continual learning frameworks, establishing it as a robust solution for continual WSI analysis in clinical applications.