Denoising diffusion models are a novel class of generative algorithms that achieve state-of-the-art performance across a range of domains, including image generation and text-to-image tasks. Building on this success, diffusion models have recently been extended to the Riemannian manifold setting, broadening their applicability to a range of problems from the natural and engineering sciences. However, these Riemannian diffusion models are built on the assumption that their forward and backward processes are well-defined for all times, preventing them from being applied to an important set of tasks that consider manifolds defined via a set of inequality constraints. In this work, we introduce a principled framework to bridge this gap. We present two distinct noising processes based on (i) the logarithmic barrier metric and (ii) the reflected Brownian motion induced by the constraints. As existing diffusion model techniques cannot be applied in this setting, we derive new tools to define such models in our framework. We then demonstrate the practical utility of our methods on a number of synthetic and real-world tasks, including applications from robotics and protein design.