We present a study of the mass-metallicity (M-Z) relation and H II region physical conditions in a sample of 20 star-forming galaxies at 1.0 < z < 1.5 drawn from the DEEP2 Galaxy Redshift Survey. We find a correlation between stellar mass and gas-phase oxygen abundance in the sample and compare it with the one observed among UV-selected z ~ 2 star-forming galaxies and local objects from the Sloan Digital Sky Survey (SDSS). This comparison, based on the same empirical abundance indicator, demonstrates that the zero point of the M-Z relationship evolves with redshift, in the sense that galaxies at fixed stellar mass become more metal-rich at lower redshift. Measurements of [O III]/Hβ and [N II]/Hα emission-line ratios show that, on average, objects in the DEEP2 1.0 < z < 1.5 sample are significantly offset from the excitation sequence observed in nearby H II regions and SDSS emission-line galaxies. In order to fully understand the causes of this offset, which is also observed in z ~ 2 star-forming galaxies, we examine in detail the small fraction of SDSS galaxies that have similar diagnostic ratios to those of the DEEP2 sample. Some of these galaxies indicate evidence for AGN and/or shock activity, which may give rise to their unusual line ratios and contribute to Balmer emission lines at the level of ~20%. Others indicate no evidence for AGN or shock excitation yet are characterized by higher electron densities and temperatures, and therefore interstellar gas pressures, than typical SDSS star-forming galaxies of similar stellar mass. These anomalous objects also have higher concentrations and starformation rate surface densities, which are directly connected to higher interstellar pressure. Higher star formation rate surface densities, interstellar pressures, and H II region ionization parameters may also be common at high redshift. These effects must be taken into account when using strong-line indicators to understand the evolution of heavy elements in galaxies. When such effects are included, the inferred evolution of the M-Z relation out to z ~ 2 is more significant than previous estimates.