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.
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