Survival prediction of cancers is crucial for clinical practice, as it informs mortality risks and influences treatment plans. However, a $\textit{static}$ model trained on a single dataset fails to adapt to the $\textit{dynamically evolving}$ 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 $\textit{catastrophic forgetting}$ and $\textit{complex inter-modal interactions}$ between gigapixel whole slide images and genomics, we propose $\textbf{ConSurv}$, the $\textbf{first}$ 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, as well as 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. Our code is provided in the supplementary material and will be made publicly available upon publication.