  We present VLT/SINFONI near-infrared (NIR) integral field spectroscopy of six
$z \sim 0.2$ Lyman break galaxy "analogs" (LBAs), from which we detect HI, HeI,
and [FeII] recombination lines, and multiple H$_2$ ro-vibrational lines in
emission. Pa$\alpha$ kinematics reveal high velocity dispersions and low
rotational velocities relative to random motions ($\langle v/\sigma \rangle =
1.2 \pm 0.8$). Matched-aperture comparisons of H$\beta$, H$\alpha$, and
Pa$\alpha$ reveal that the nebular color excesses are lower relative to the
continuum color excesses than is the case for typical local star-forming
systems. We compare observed HeI/HI recombination line ratios to
photoionization models to gauge the effective temperatures (T$_{\rm eff}$) of
massive ionizing stars, finding the properties of at least one LBA are
consistent with extra heating from an active galactic nucleus (AGN) and/or an
overabundance of massive stars. We use H$_2$ 1-0 S($\cdot$) ro-vibrational
spectra to determine rotational excitation temperature $T_{\rm ex} \sim 2000$ K
for warm molecular gas, which we attribute to UV heating in dense
photon-dominated regions. Spatially resolved NIR line ratios favor excitation
by massive, young stars, rather than supernovae or AGN feedback. Our results
suggest that the local analogs of Lyman break galaxies are primarily subject to
strong feedback from recent star formation, with evidence for AGN and outflows
in some cases.
