There is enormous recent interest in weak, van der Waals-type (vdW) interactions due to their fundamental relevance for two-dimensional materials and the so-called vdW heterostructures. Tackling this problem using computer simulation is very challenging due to the nontrivial, nonlocal nature of these interactions. We benchmark different treatments of London dispersion forces within the density functional theory (DFT) framework on hexagonal boron nitride or graphene monolayers on Ir(111) by comparing the calculated geometries to a comprehensive set of experimental data. The geometry of these systems crucially depends on the interplay between vdW interactions and wave function hybridization, making them excellent test cases for vdW-treated DFT. Our results show strong variations in the calculated atomic geometry. While some of the approximations reproduce the experimental structure, this is rather based on a posteriori comparison with the "target results." General predictive power in vdW-treated DFT is not achieved yet and might require new approaches.