Context. Post-asymptotic giant branch (AGB) binaries are surrounded by circumbinary disks of gas and dust that are similar to protoplanetary disks found around young stars. Aims. We aim to understand the structure of these disks and identify the physical phenomena at play in their very inner regions. We want to understand the disk-binary interaction and to further investigate the comparison with protoplanetary disks. Methods. We conducted an interferometric snapshot survey of 23 post-AGB binaries in the near-infrared (H-band) using VLTI/PIONIER. We fit the multi-wavelength visibilities and closure phases with purely geometrical models with an increasing complexity (including two point-sources, an azimuthally modulated ring, and an over-resolved flux) in order to retrieve the sizes, temperatures, and flux ratios of the different components. Results. All sources are resolved and the different components contributing to the H-band flux are dissected. The environment of these targets is very complex: 13/23 targets need models with thirteen or more parameters to fit the data. We find that the inner disk rims follow and extend the size-luminosity relation established for disks around young stars with an offset toward larger sizes. The measured temperature of the near-infrared circumstellar emission of post-AGB binaries is lower (T_sub ~ 1200 K) than for young stars, which is probably due to a different dust mineralogy and/or gas density in the dust sublimation region. Conclusions. The dusty inner rims of the circumbinary disks around post-AGB binaries are ruled by dust sublimation physics. Additionally a significant amount of the circumstellar H-band flux is over-resolved (more than 10% of the non-stellar flux is over-resolved in 14 targets). This hints that a source of unknown origin, either a disk structure or outflow. The amount of over-resolved flux is larger than around young stars. Due to the complexity of these targets, interferometric imaging is a necessary tool to reveal the interacting inner regions in a model-independent way.