We analyze the rest-optical emission-line ratios of z~1.5 galaxies drawn from the MOSFIRE Deep Evolution Field (MOSDEF) survey. Using composite spectra we investigate the mass-metallicity relation (MZR) at z~1.5 and measure its evolution to z=0. When using gas-phase metallicities based on the N2 line ratio, we find that the MZR evolution from z~1.5 to z=0 depends on stellar mass, evolving by $Deltarm log(rm O/H)sim0.25$ dex at $M_*<10^{9.75}M_{odot}$ down to $Deltarm log(rm O/H)sim0.05$ at $M_*>10^{10.5}M_{odot}$. In contrast, the O3N2-based MZR shows a constant offset of $Deltarm log(rm O/H)sim0.30$ across all masses, consistent with previous MOSDEF results based on independent metallicity indicators, and suggesting that O3N2 provides a more robust metallicity calibration for our z~1.5 sample. We investigated the secondary dependence of the MZR on SFR by measuring correlated scatter about the mean $M_*$-specific SFR and $M_*-log(rm O3N2)$ relations. We find an anti-correlation between $log(rm O/H)$ and sSFR offsets, indicating the presence of a $M_*$-SFR-Z relation, though with limited significance. Additionally, we find that our z~1.5 stacks lie along the z=0 metallicity sequence at fixed $mu=log(M_*/M_{odot})-0.6timeslog(rm SFR / M_{odot} yr^{-1})$ suggesting that the z~1.5 stacks can be described by the z=0 fundamental metallicity relation (FMR). However, using different calibrations can shift the calculated metallicities off of the local FMR, indicating that appropriate calibrations are essential for understanding metallicity evolution with redshift. Finally, understanding how [NII]/H$alpha$ scales with galaxy properties is crucial to accurately describe the effects of blended [NII] and H$alpha$ on redshift and H$alpha$ flux measurements in future large surveys utilizing low-resolution spectra such as with Euclid and the Roman Space Telescope.
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