Hey,
AFAIK, as long as Stan/HMC/NUTS samples efficiently you should not have to be worried more than usually. However, I’m not 100% sure about that!
I’d also presume that one could come up with some example where Stan’s diagnostics do not complain (high average acceptance rate, no divergences, low rhat, high ESS etc), but your inference is “invalid” (e.g. biased) due in part to the inexact gradients.
Maybe this thread and the referenced linked therein can help you as well: What (exactly) happens for only piecewise continuous posterior densities, gradients or wrong gradients?