Language models trained for clinical disease inference are trained on patient data, which may include sensitive and private information, and data owners may request the removal of their data from a trained model due to privacy or copyright concerns. However, exactly unlearning patient-specific data is intractable, and retraining with minor data removal is resource-intensive. While there exists several machine unlearning methods that can be used, their utility is generally restricted to non-medical domains. Moreover, the existing benchmarks for evaluating such unlearning methods primarily utilize synthetically curated datasets, which are not truly representative of real-world systems. Hence, the effectiveness of these unlearning methods in the medical domain is largely unclear. To this end, we introduce REMEDI, an extensive benchmark for machine unlearning tailored to multi-label and multiclass clinical disease inference, where label correlations, longitudinal structure, and safety constraints make unlearning particularly challenging. Unlike the existing benchmarks, REMEDI considers: (1) a relevant application domain (medical), (2) comprehensive unlearning setups involving diverse sets of forget instances, (3) challenging unlearning scenarios including multi-label and multi-class classification tasks, and (4) evaluation metrics involving performance both in terms of utility and extent of unlearning achieved. REMEDI is developed using the MIMIC-III clinical database that contains comprehensive clinical data of patients. Experiments with existing unlearning methods indicate that there exists a trade-off between utility and unlearning performance. They are also largely unsuited to multi-label classification tasks. To facilitate reproducibility, we make our benchmark publicly available.