  We demonstrate that fermion-boson models with nonlocal interactions can be
simulated at finite band filling with the continuous-time quantum Monte Carlo
method. We apply this method to explore the influence of the electron-phonon
interaction range for a half-filled band in one dimension, covering the full
range from the Holstein to the Fr\"ohlich regime. The phase diagram contains
metallic, Peierls, and phase-separated regions, which we characterize in terms
of static and dynamical correlation functions. In particular, our results
reveal a suppression of $2k_F$ charge correlations with increasing interaction
range, allowing for a power-law decay comparable to the pairing correlations.
