There are currently three main classes of liquid-repellent surfaces: micro- or nanostructured superhydrophobic surfaces, flat surfaces grafted with "liquidlike" polymer brushes, and lubricated surfaces. Despite recent progress, the mechanistic explanation for the differences in droplet behavior on such surfaces is still under debate. Here, we measure the dissipative force acting on a droplet moving on representatives of these surfaces at different velocities U=0.01-1 mm/s using a cantilever force sensor with submicronewton accuracy and correlate it to the contact line dynamics observed using optical interferometry at high spatial (micron) and temporal (<0.1 s) resolutions. We find that the dissipative force - due to very different physical mechanisms at the contact line - is independent of velocity on superhydrophobic surfaces but depends nonlinearly on velocity for flat and lubricated surfaces. The techniques and insights presented here will inform future work on liquid-repellent surfaces and enable their rational design.