Suppose my real model has parameters \beta but I do a QR decomposition and estimate \theta for the transformed data. What if I want to put a prior on \beta? The manual indicates that priors aren’t so important with the QR decomposition, so maybe this is a silly concern. But I know the values of beta are smallish, and would like to incorporate that info.

If there is any point in putting the priors on \beta, do I need to worry about the Jacobean?

You don’t have to worry about the Jacobian. Either you are putting the prior on the primitive \boldsymbol{\theta} (which is what rstanarm does) or you are putting the prior on the transformed parameter \boldsymbol{\beta}, in which case the Jacobian is constant because it only involves the known \mathbf{R}.

Since \boldsymbol{\beta} = \mathbf{R}^{-1} \boldsymbol{\theta}, if you believe \boldsymbol{\beta} tends to be close to \mathbf{0}, then the same is true about \boldsymbol{\theta}, although there is a re-scaling going on. Basically, something weakly informative with a center of \mathbf{0} should be fine either way.