The ([Fe II] 1.644 mu m)/Br gamma and ([Fe II] 1.2567 mu m)/Pa beta ratios provide good separation of starburst and Seyfert galaxies; we show that ([Fe II] 1.644 mu m)/Br gamma is nearly as good a discriminant as the commonly used ([O I] lambda 6300)/H alpha ratio. The behavior of both the ([O I] lambda 6300)/H alpha and the ([Fe II] 1.644 mu m)/Br gamma ratios is readily understood if there is a progression in the proportion of shock excitation going from H II region through starburst and Seyfert galaxies to supernova remnants. We use the CLOUDY program to show that the behavior is also explained if the dominant excitation mechanism is photoionization. As a test of this technique for identifying excitation mechanisms, we have obtained infrared spectra of two LINER galaxies. NGC 1052, which is thought to be excited by a nuclear power-law UV source, is the prototype of LINERs, and NGC 3504 is classified as a weak-[O I] lambda 6300 LINER, better understood in terms of photoionization by O stars than by a nonstellar source. The placement of these galaxies on the ([Fe II] 1.644 mu m)/Br gamma versus ([O I] lambda 6300)/H alpha diagram is in agreement with these previously determined characteristics. The ([Fe II] 1.644 mu m)/Br gamma and ([Fe II] 1.2567 mu m)/Pa beta ratios should be useful in classifying heavily obscured galaxies because the ratios (1) are constructed at wavelengths where extinction is relatively small and (2) involve lines over narrow spectral ranges.