  The unprecedented infrared spectroscopic capabilities of JWST have provided
high-quality interstellar medium (ISM) metallicity measurements and enabled
characterization of the gas-phase mass-metallicity relation (MZR) for galaxies
at $z \gtrsim 5$ for the first time. We analyze the gas-phase MZR and its
evolution in a high-redshift suite of FIRE-2 cosmological zoom-in simulations
at $z=5-12$ and for stellar masses $M_* \sim 10^6-10^{10} \rm{M}_\odot$. These
simulations implement a multi-channel stellar feedback model and produce
broadly realistic galaxy properties, including when evolved to $z=0$. The
simulations predict very weak redshift evolution of the MZR over the redshift
range studied, with the normalization of the MZR increasing by less than $0.01$
dex as redshift decreases from $z = 12$ to $z=5$. The median MZR in the
simulations is well-approximated as a constant power-law relation across this
redshift range given by $\log(Z/Z_\odot) = 0.37\log(M_*/\rm{M}_\odot) - 4.3$.
We find good agreement between our best-fit model and recent observations made
by JWST at high redshift. The weak evolution of the MZR at $z > 5$ contrasts
with the evolution at $z \lesssim 3$, where increasing normalization of the MZR
with decreasing redshift is observed and predicted by most models. The FIRE-2
simulations predict increasing scatter in the gas-phase MZR with decreasing
stellar mass, in qualitative agreement with some observations.
