Lithium abundances were measured in 17 subgiants and giants which have evolved from the main-sequence Li dip in the old open cluster M67. The absence of detectable Li in all but one subgiant argues strongly that Li is severely destroyed, not merely diffused from observable view, in the main-sequence stars. A comparison with the Hyades, Praesepe NGC 752, and M67 clusters is performed, requiring the adoption of a uniform temperature calibration, the use of newly standardized photometry for NGC 752, and the rederivation of Li abundances from published equivalent widths. Limitations of the commonly used temperature calibrations, and our adopted choice of the Saxner & Hammarbäck (1985) calibration, are discussed. Adoption of this calibration results in a significant change in the shape of the Hyades dip.A comparison of the Li dips in the various clusters reveals intriguing details. The mass at the Li dip depends upon the metallicity of the stars, but the zero-age main-sequence (ZAMS) temperatures of the dip stars are found to be independent of their metallicity. The morphology of the Li dip is characterized by a sharp drop at the blue edge and a more gradual rise at the red edge. There is no change in the ZAMS temperature or shape of the blue edge with age. The red edge becomes less steep with age; this may either be due to the evolution of the Li dip, or be caused by a decrease in Li in stars cooler than the red edge and thus unrelated to the Li dip phenomenon. The Li versus effective temperature distributions in the Hyades and Praesepe clusters, which are of the same age but different metallicity, are found to be identical in both the F and G dwarfs. None of the models so far proposed adequately explains the data. Microscopic diffusion and mass loss are contradicted by the observations of the Li dip. Models which incorporate rotation produce the Li dip through meridional circulation, turbulence, or rotational braking. Meridional circulation is in conflict with observations of Li outside the Li dip. All of the rotation models would produce a larger scatter in the Li dip than is observed. Observations suggest that the stars in the Li dip may undergo spin-down on a much longer timescale than late-F and G dwarfs. Examination of the rotational velocity distribution in dip stars in a cluster older than the Hyades may provide a clue to the anomalous mixing and Li depletion in the dip stars.