Central to the adhesion and locomotion of tree frogs are their structured toe pads, which consist of an array of 10 mu m hexagonal epithelial cells separated by interconnected channels that are 1 mu m wide and 10 mu m deep. It has been proposed that the channels facilitate the drainage of excess fluid trapped between the toe pads and the contacting surface, and thus reduce the hydrodynamic repulsion during approach. We performed direct force measurement of the normal hydrodynamic interactions during the drainage of fluid from the gap between a structured and a smooth surface using surface force apparatus. The structured surface consisted of a hexagonal array of cylindrical posts to represent the network of interconnected channels. The measured hydrodynamic drainage forces agree with the predictions from Reynolds' theory for smooth surfaces at large separations. Deviations from theory, characterized by a reduction in the hydrodynamic repulsion, are observed below some critical separation (h(c)), which is independent of drive velocity. We employ a scaling analysis to establish the relationship between structural features (channel depth, width, and post diameter) and the critical separation for the onset of deviations. We find agreement between our experiments and the scaling analysis, which allows us to estimate a characteristic length scale that corresponds to the transition from the fluid being radially squeezed out of the nominal contact area to being squeezed out through the network of interconnected channels.