Clinical deployment of chest radiograph classifiers requires models that can be updated as new datasets become available without retraining on previously observed data or degrading validated performance. We study a task-incremental continual learning setting for chest radiograph classification under task-unknown inference, where heterogeneous chest X-ray datasets arrive sequentially and task identity is unavailable at deployment time. We propose CARL-CXR, a continual adapter-based routing framework that maintains a fixed high-capacity backbone while incrementally introducing lightweight task-specific adapters and classifier heads. A latent task selector operates on adapter-conditioned features to dynamically route each input to the most relevant task pathway, leveraging compact task prototypes and feature-level experience replay to preserve task identity across sequential updates without storing raw images. Experiments on MIMIC-CXR and CheXpert two large-scale datasets with distinct patient populations, imaging devices, and annotation pipelines demonstrate that CARL-CXR achieves minimal catastrophic forgetting (0.012 AUROC drop), representing a 6X and 11X reduction over established continual learning baselines LwF and EWC respectively, while maintaining competitive diagnostic performance (AUROC 0.74). Under task unknown deployment, CARL-CXR outperforms joint training by 12.5 points in routing accuracy (75.0% vs. 62.5%): unlike LwF and EWC, which require explicit task identifiers at inference and provide no routing mechanism.