  We present detections of [OIII]$\lambda$4363 and direct-method metallicities
for star-forming galaxies at $z=1.7-3.6$. We combine new measurements from the
MOSFIRE Deep Evolution Field (MOSDEF) survey with literature sources to
construct a sample of 18 galaxies with direct-method metallicities at $z>1$,
spanning $7.5<1$2+log(O/H$)<8.2$ and log(M$_*$/M$_{\odot})=7-10$. We find that
strong-line calibrations based on local analogs of high-redshift galaxies
reliably reproduce the metallicity of the $z>1$ sample on average. We construct
the first mass-metallicity relation at $z>1$ based purely on direct-method O/H,
finding a slope that is consistent with strong-line results. Direct-method O/H
evolves by $\lesssim0.1$ dex at fixed M$_*$ and SFR from $z\sim0-2.2$. We
employ photoionization models to constrain the ionization parameter and
ionizing spectrum in the high-redshift sample. Stellar models with super-solar
O/Fe and binary evolution of massive stars are required to reproduce the
observed strong-line ratios. We find that the $z>1$ sample falls on the
$z\sim0$ relation between ionization parameter and O/H, suggesting no evolution
of this relation from $z\sim0$ to $z\sim2$. These results suggest that the
offset of the strong-line ratios of this sample from local excitation sequences
is driven primarily by a harder ionizing spectrum at fixed nebular metallicity
compared to what is typical at $z\sim0$, naturally explained by super-solar
O/Fe at high redshift caused by rapid formation timescales. Given the extreme
nature of our $z>1$ sample, the implications for representative $z\sim2$ galaxy
samples at $\sim10^{10}$ M$_{\odot}$ are unclear, but similarities to $z>6$
galaxies suggest that these conclusions can be extended to galaxies in the
epoch of reionization.
