Survival prediction of cancers is crucial for clinical practice, as it informs mortality risks and influences treatment plans. However, a static model trained on a single dataset fails to adapt to the dynamically evolving clinical environment and continuous data streams, limiting its practical utility. While continual learning (CL) offers a solution to learn dynamically from new datasets, existing CL methods primarily focus on unimodal inputs and suffer from severe catastrophic forgetting in survival prediction. In real-world scenarios, multimodal inputs often provide comprehensive and complementary information, such as whole slide images and genomics; and neglecting inter-modal correlations negatively impacts the performance. To address the two challenges of catastrophic forgetting and complex inter-modal interactions between gigapixel whole slide images and genomics, we propose ConSurv, the first multimodal continual learning (MMCL) method for survival analysis. ConSurv incorporates two key components: Multi-staged Mixture of Experts (MS-MoE) and Feature Constrained Replay (FCR). MS-MoE captures both task-shared and task-specific knowledge at different learning stages of the network, including two modality encoders and the modality fusion component, learning inter-modal relationships. FCR further enhances learned knowledge and mitigates forgetting by restricting feature deviation of previous data at different levels, including encoder-level features of two modalities and the fusion-level representations. Additionally, we introduce a new benchmark integrating four datasets, Multimodal Survival Analysis Incremental Learning (MSAIL), for comprehensive evaluation in the CL setting. Extensive experiments demonstrate that ConSurv outperforms competing methods across multiple metrics.