Diffusion models have recently emerged as powerful generative models in medical imaging. However, it remains a major challenge to combine these data-driven models with domain knowledge to guide brain imaging problems. In neuroimaging, Bayesian inverse problems have long provided a successful framework for inference tasks, where incorporating domain knowledge of the imaging process enables robust performance without requiring extensive training data. However, the anatomical modeling component of these approaches typically relies on classical mathematical priors that often fail to capture the complex structure of brain anatomy. In this work, we present the first general-purpose application of diffusion models as priors for solving a wide range of medical imaging inverse problems. Our approach leverages a score-based diffusion prior trained extensively on diverse brain MRI data, paired with flexible forward models that capture common image processing tasks such as super-resolution, bias field correction, inpainting, and combinations thereof. We further demonstrate how our framework can refine outputs from existing deep learning methods to improve anatomical fidelity. Experiments on heterogeneous clinical and research MRI data show that our method achieves state-of-the-art performance producing consistent, high-quality solutions without requiring paired training datasets. These results highlight the potential of diffusion priors as versatile tools for brain MRI analysis.