© 2015. The American Astronomical Society. All rights reserved. We present results from the MOSFIRE Deep Evolution Field (MOSDEF) survey on rest-frame optical active galactic nucleus (AGN) identification and completeness at z ∼ 2.3. With our sample of 50 galaxies and 10 X-ray and IR-selected AGNs with measured Hβ, [O III], Hα, and N II emission lines, we investigate the location of AGNs in the BPT, MEx (mass-excitation), and CEx (color-excitation) diagrams. We find that th BPT diagram works well to identify AGNs at z ∼ 2.3 and that the z ∼ 0 AGN/star-forming galaxy classifications do not need to shift substantially at z ∼ 2.3 to robustly separate these populations. However, the MEx diagram fails to identify all of the AGN identified in the BPT diagram, and the CEx diagram is substantially contaminated at high redshift. We further show that AGN samples selected using the BPT diagram have selection biases in terms of both host stellar mass and stellar population, in that AGNs in low mass and/or high specific star formation rate galaxies are difficult to identify using the BPT diagram. These selection biases become increasingly severe at high redshift, such that optically selected AGN samples at high redshift will necessarily be incomplete. We also find that the gas in the narrow-line region appears to be more enriched than gas in the host galaxy for at least some MOSDEF AGNs. However, AGNs at z ∼ 2 are generally less enriched than local AGNs with the same host stellar mass.