A photoionization analysis of Hubble Space Telescope UV and ground-based optical spectrophotometry is given for eight more planetary nebulae (PNs) in the Large Magellanic Cloud (LMC). This allows the central stars to be placed accurately on the H-R diagram and permits the determination of the He, C, N, O, Ne, S, and Ar abundances. In some cases, the gas-phase abundances of Mg and Si may also be determined. We have combined these results with the analysis of two other objects published by us in the first two papers of this series. The observed abundance patterns are qualitatively consistent with the (mass-dependent) operation of the various chemical dredge-up processes as predicted by theory. Dredge-up of C during the thermal pulsing stage appears to be most important, and "hot bottom burning" transforms much of this C to N in the more massive stars. There is no sign of dredge-up of 22Ne. We show that the spread in the alpha -process element abundances can be understood as being due to differences in core mass of the planetary nebula nucleus, which is related directly to initial mass of the precursor star. This is, therefore, a tracer of the age-metallicity relationship for stars in general, and we derive, for the first time, the chemical history of the LMC based on PNs. We find that the base metallicity of the LMC almost doubled ~2 Gyr ago. This is consistent with studies of field stars and of clusters that show that there was a major burst of star formation at that time.