  We present James Webb Space Telescope (JWST) Integral Field Spectrograph
observations of NGC 3256, a local infrared-luminous late-stage merging system
with two nuclei about 1 kpc apart, both of which have evidence of cold
molecular outflows. Using JWST NIRSpec and MIRI datasets, we investigate this
morphologically complex system on spatial scales of $<$100 pc, where we focus
on the warm molecular H$_2$ gas surrounding the nuclei. We detect collimated
outflowing warm H$_2$ gas originating from the southern nucleus, though we do
not find significant outflowing warm H$_2$ gas surrounding the northern
nucleus. Within the observed region, the maximum intrinsic velocities of the
outflow reach up to $\sim$1,000 km s$^{-1}$, and extend out to a distance of
0.7 kpc. Based on H$_2$ S(7)/S(1) line ratios, we find a larger fraction of
warmer gas near the S nucleus, which decreases with increasing distance from
the nucleus, signifying the S nucleus as a primary source of H$_2$ heating. The
gas mass of the warm H$_2$ outflow component is estimated to be
$M\rm{_{warm,out}}$ = 8.9$\times$10$^5\;M_{\odot}$, as much as 4$\%$ of the
cold H$_2$ mass as estimated using ALMA CO data. The outflow time scale is
about $7\times10^5$ yr, resulting in a mass outflow rate of
$\dot{M}\rm{_{warm,out}}$ = 1.3 M$_{\odot}$ yr$^{-1}$ and kinetic power of
$P\rm{_{warm,out}}\;\sim\;2\times10^{41}$ erg s$^{-1}$. Lastly, the regions
where the outflowing gas reside show high [FeII]/Pa$\beta$ and H$_2$/Br$\gamma$
line ratios, indicating enhanced mechanical heating caused by the outflows. At
the same time, the 3.3 $\mu$m and 6.2 $\mu$m Polycyclic Aromatic Hydrocarbon
fluxes in these regions are not significantly suppressed compared to those
outside the outflows, suggesting the outflows have no clear negative feedback
effect on the local star formation.
