We present evolutionary models of subgiant stars in M67 that follow the surface abundances of lithium and beryllium. These models include a number of physical processes that have been proposed as explanations for the main-sequence depletion of lithium in the Hyades. We use the observed lithium abundance as a function of temperature in M67 subgiants and the correlation between lithium and beryllium depletion to distinguish between these models. We find that constant main-sequence mass loss and diffusion are strongly ruled out as viable mechanisms for light-element depletion in main-sequence stars. Rotationally induced mixing is strongly supported by the lithium- and beryllium-depletion correlation observed in Population I field F stars; such mixing is also consistent with the lithium abundances observed in M67 subgiants, although we discuss some possible complications.