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We investigate the evolution of galaxy gas-phase metallicity (O/H) over the range $z=0-3.3$ using samples of $\sim300$ galaxies at $z\sim2.3$ and $\sim150$ galaxies at $z\sim3.3$ from the MOSDEF survey. This analysis crucially utilizes different metallicity calibrations at $z\sim0$ and $z>1$ to account for evolving ISM conditions. We find significant correlations between O/H and stellar mass ($M_*$) at $z\sim2.3$ and $z\sim3.3$. The low-mass power law slope of the mass-metallicity relation is remarkably invariant over $z=0-3.3$, such that $\textrm{O/H}\propto M_*^{0.30}$ at all redshifts in this range. At fixed $M_*$, O/H decreases with increasing redshift as dlog(O/H)/d$z=-0.11\pm0.02$. We find no evidence that the fundamental metallicity relation between $M_*$, O/H, and star-formation rate (SFR) evolves out to $z\sim3.3$, with galaxies at $z\sim2.3-3.3$ having O/H within 0.04~dex of local galaxies matched in $M_*$ and SFR on average. We employ analytic chemical evolution models to place constraints on the mass and metal loading factors of galactic outflows. The efficiency of metal removal increases toward lower $M_*$ at fixed redshift, and toward higher redshift at fixed $M_*$. These models suggest that the slope of the mass-metallicity relation is set by the scaling of the metal loading factor of outflows with $M_*$, not by the change in gas fraction as a function of $M_*$. The evolution toward lower O/H at fixed $M_*$ with increasing redshift is driven by both higher gas fraction (leading to stronger dilution of ISM metals) and higher metal removal efficiency, with models suggesting that both effects contribute approximately equally to the observed evolution. These results suggest that the processes governing the smooth baryonic growth of galaxies via gas flows and star formation hold in the same form over at least the past 12~Gyr.