© 2018. The American Astronomical Society. All rights reserved.. We combine Hα and Hβ spectroscopic measurements and UV photometry for a sample of 673 galaxies from the MOSDEF survey to constrain hydrogen-ionizing photon production efficiencies (ζion) at z = 1.4-2.6. We find 〈log(ζion/[s-1erg s-1Hz-1])〉=25.06(25.34), assuming the Calzetti (SMC) curve for the UV dust correction and a scatter of 0.28 dex in the ζiondistribution. After accounting for observational uncertainties and variations in dust attenuation, we conclude that the remaining scatter in ζionis likely dominated by galaxy-to-galaxy variations in stellar populations, including the slope and upper-mass cutoff of the initial mass function, stellar metallicity, star formation burstiness, and stellar evolution (e.g., single/binary star evolution). Moreover, ζionis elevated in galaxies with high ionization states (high [O iii]/[O ii]) and low oxygen abundances (low [N ii]/Hα and high [O iii]/Hβ) in the ionized ISM. However, ζiondoes not correlate with the offset from the z ∼ 0 star-forming locus in the BPT diagram, suggesting no change in the hardness of the ionizing radiation accompanying the offset from the z ∼ 0 sequence. We also find that galaxies with blue UV spectral slopes (〈β〉=-2.1) have ζionelevated by a factor of ∼2 relative to the average ζionof the sample (〈β〉=-1.4). If these blue galaxies are similar to those at z > 6, our results suggest that a lower Lyman-continuum escape fraction is required for galaxies to maintain reionization, compared to the canonical predictions from stellar population models. Furthermore, we demonstrate that even with robustly dust-corrected Hα, the UV dust attenuation can cause on average a ∼0.3 dex systematic uncertainty in calculations.