Near-infrared spectra of 15 high-redshift radio galaxies (HzRGs) located at 2 < z < 2.6 were obtained by the OH Airglow Suppressor spectrograph mounted on the Subaru telescope. The UV-optical line ratio diagnostic diagrams indicate that half of the observed HzRGs have extended emission-line regions with low metal abundance, photoionized by a flat-continuum active galactic nucleus such as a quasar. We also found two probable correlations between radio and rest-optical parameters: (1) HzRGs with massive hosts tend to have a redder rest-optical continuum, and (2) HzRGs with smaller radio sizes also show a redder optical continuum. On the basis of the correlations, the nature of HzRGs at 2 < z < 2.6 is discussed.