We explore the kinematics of ionized gas via the [O III] $lambda$5007 emission lines in active galactic nuclei (AGN) selected on the basis of their mid-infrared (IR) emission, and split into obscured and unobscured populations based on their optical-IR colors. After correcting for differences in redshift distributions, we provide composite spectra of spectroscopically and photometrically defined obscured/Type 2 and unobscured/Type 1 AGN from 3500 to 7000 AA. The IR-selected obscured sources contain a mixture of narrow-lined Type 2 AGN and intermediate sources that have broad H$alpha$ emission and significantly narrower H$beta$. Using both [OIII] luminosities and AGN luminosities derived from optical-IR spectral energy distribution fitting, we find evidence for enhanced large-scale obscuration in the obscured sources. In matched bins of luminosity we find that the obscured population typically has broader, more blueshifted OIII emission than in the unobscured sample, suggestive of more powerful AGN-driven outflows. This trend is not seen in spectroscopically classified samples, and is unlikely to be entirely explained by orientation effects. In addition, outflow velocities increase from small to moderate AGN $E(B-V)$ values, before flattening out (as traced by FWHM) and even decreasing (as traced by blueshift). While difficult to fully interpret in a single physical model, due to both the averaging over populations and the spatially-averaged spectra, these results agree with previous findings that simple geometric unification models are insufficient for the IR-selected AGN population, and may fit into an evolutionary model for obscured and unobscured AGN.