We have used FMOS on Subaru to obtain near-infrared spectroscopy of 123 far-infrared-selected galaxies in COSMOS and the key rest-frame optical emission lines. This is the largest sample of infrared galaxies with near-infrared spectroscopy at these redshifts. The far-infrared selection results in a sample of galaxies that are massive systems that span a range of metallicities in comparison with previous optically selected surveys, and thus has a higher active galactic nucleus (AGN) fraction and better samples the AGN branch. We establish the presence of AGNs and starbursts in this sample of (U)LIRGs selected as Herschel-PACS and Spitzer-MIPS detections in two redshift bins (z ~ 0.7 and z ~ 1.5) and test the redshift dependence of diagnostics used to separate AGNs from star formation dominated galaxies. In addition, we construct a low-redshift (z ~ 0.1) comparison sample of infrared-selected galaxies and find that the evolution from z ~ 1.5 to today is consistent with an evolving AGN selection line and a range of ISM conditions and metallicities from the models of Kewley et al. We find that a large fraction of (U)LIRGs are BPT-selected AGNs using their new redshift-dependent classification line. We compare the position of known X-ray-detected AGNs (67 in total) with the BPT selection and find that the new classification line accurately selects most of these objects (>70%). Furthermore, we identify 35 new (likely obscured) AGNs not selected as such by their X-ray emission. Our results have direct implications for AGN selection at higher redshift with either current (MOSFIRE, KMOS) or future (PFS, MOONS) spectroscopic efforts with near-infrared spectral coverage.