Using observations from the MOSFIRE Deep Evolution Field survey, we investigate the physical conditions of star-forming regions in z ~ 2.3 galaxies, specifically the electron density and ionization state. From measurements of the OIIlambdalambda3726,3729 and SIIlambdalambda6716,6731 doublets, we find a median electron density of ~250 cm super(-3) at z ~ 2.3, an increase of an order of magnitude compared to measurements of galaxies at z ~ 0. While z ~ 2.3 galaxies are offset toward significantly higher O sub(32) values relative to local galaxies at fixed stellar mass, we find that the high-redshift sample follows a similar distribution to the low-metallicity tail of the local distribution in the O sub(32) versus R sub(23) and O3N2 diagrams. Based on these results, we propose that z ~ 2.3 star-forming galaxies have the same ionization parameter as local galaxies at fixed metallicity. In combination with simple photoionization models, the position of local and z ~ 2.3 galaxies in excitation diagrams suggests that there is no significant change in the hardness of the ionizing spectrum at fixed metallicity from z ~ 0 to z ~ 2.3. We find that z ~ 2.3 galaxies show no offset compared to low-metallicity local galaxies in emission line ratio diagrams involving only lines of hydrogen, oxygen, and sulfur, but show a systematic offset in diagrams involving NIIlambda6584. We conclude that the offset of z ~ 2.3 galaxies from the local star-forming sequence in the NII BPT diagram is primarily driven by elevated N/O at fixed O/H compared to local galaxies. These results suggest that the local gas-phase and stellar metallicity sets the ionization state of star-forming regions at z ~ 0 and z ~ 2.