Room impulse response (RIR) estimation naturally arises as a class of inverse problems, including denoising and deconvolution. While recent approaches often rely on supervised learning or learned generative priors, such methods require large amounts of training data and may generalize poorly outside the training distribution. In this work, we present RIRFlow, a training-free Bayesian framework for RIR inverse problems using flow matching. We derive a flow-consistent analytic prior from the statistical structure of RIRs, eliminating the need for data-driven priors. Specifically, we model RIR as a Gaussian process with exponentially decaying variance, which yields a closed-form minimum mean squared error (MMSE) Wiener denoiser. This analytic denoiser is integrated as a prior in an existing flow-based inverse solver, where inverse problems are solved via guided posterior sampling. Furthermore, we extend the solver to nonlinear and non-Gaussian inverse problems via a local Gaussian approximation of the guided posterior, and empirically demonstrate that this approximation remains effective in practice. Experiments on real RIRs across different inverse problems demonstrate robust performance, highlighting the effectiveness of combining a classic RIR model with the recent flow-based generative inference.