Predictive machine learning models are becoming increasingly deployed in high-stakes contexts involving sensitive personal data; in these contexts, there is a trade-off between model explainability and data privacy. In this work, we push the boundaries of this trade-off: with a focus on foundation models for image classification fine-tuning, we reveal unforeseen privacy risks of post-hoc model explanations and subsequently offer mitigation strategies for such risks. First, we construct VAR-LRT and L1/L2-LRT, two new membership inference attacks based on feature attribution explanations that are significantly more successful than existing explanation-leveraging attacks, particularly in the low false-positive rate regime that allows an adversary to identify specific training set members with confidence. Second, we find empirically that optimized differentially private fine-tuning substantially diminishes the success of the aforementioned attacks, while maintaining high model accuracy. We carry out a systematic empirical investigation of our 2 new attacks with 5 vision transformer architectures, 5 benchmark datasets, 4 state-of-the-art post-hoc explanation methods, and 4 privacy strength settings.