The implementation of deep learning models for medical image classification poses significant challenges, including gradual performance degradation and limited adaptability to new diseases. However, frequent retraining of models is unfeasible and raises concerns about healthcare privacy due to the retention of prior patient data. To address these issues, this study investigated privacy-preserving continual learning methods as an alternative solution. We evaluated twelve privacy-preserving non-storage continual learning algorithms based deep learning models for classifying retinal diseases from public optical coherence tomography (OCT) images, in a class-incremental learning scenario. The OCT dataset comprises 108,309 OCT images. Its classes include normal (47.21%), drusen (7.96%), choroidal neovascularization (CNV) (34.35%), and diabetic macular edema (DME) (10.48%). Each class consisted of 250 testing images. For continuous training, the first task involved CNV and normal classes, the second task focused on DME class, and the third task included drusen class. All selected algorithms were further experimented with different training sequence combinations. The final model's average class accuracy was measured. The performance of the joint model obtained through retraining and the original finetune model without continual learning algorithms were compared. Additionally, a publicly available medical dataset for colon cancer detection based on histology slides was selected as a proof of concept, while the CIFAR10 dataset was included as the continual learning benchmark. Among the continual learning algorithms, Brain-inspired-replay (BIR) outperformed the others in the continual learning-based classification of retinal diseases from OCT images, achieving an accuracy of 62.00% (95% confidence interval: 59.36-64.64%), with consistent top performance observed in different training sequences. For colon cancer histology classification, Efficient Feature Transformations (EFT) attained the highest accuracy of 66.82% (95% confidence interval: 64.23-69.42%). In comparison, the joint model achieved accuracies of 90.76% and 89.28%, respectively. The finetune model demonstrated catastrophic forgetting in both datasets. Although the joint retraining model exhibited superior performance, continual learning holds promise in mitigating catastrophic forgetting and facilitating continual model updates while preserving privacy in healthcare deep learning models. Thus, it presents a highly promising solution for the long-term clinical deployment of such models.