Emission-line spectroscopy in the infrared has advantages over optical methods, particularly for investigating active galaxies with nuclei obscured by dust. We predict intensities and ratios of the near-, mid-, and far-infrared ionic fine-structure lines from active galactic nuclei (AGNs) of the LINER and Seyfert types, and starbursts, using Ferland's photoionization code CLOUDY. Several infrared line ratio diagrams effectively distinguish emission from these sources and from shocks and photodissociation regions (PDRs) in the neutral interstellar medium of the host galaxy. We have studied extensively the effects of varying the important model parameters, including ionization parameter, density, and optical depth. We also show how detailed infrared spectroscopy can reveal the presence of density inhomogeneities and abundances anomalies and determine the slope of the ionizing continuum and the amount of reddening and the reddening law. Further measurements of infrared line ratios can determine whether the observed coronal emission lines in Seyfert nuclei arise from a diffuse low-density gas spread over hundreds of parsecs or from high-density gas intermediate in location and properties between the broad- and narrow-line regions. As a test, we compared the predictions of our model and others to the observed line ratios of the starburst galaxy M82 and of the Galactic center. No single-zone model adequately predicted all of the observed lines, but a simple combination of a starburst plus photodissociation region of higher density gave a good fit for M82. Finally, we use our model predictions to estimate the detectability of infrared fine-structure lines in various active galaxies, using current and planned ground, airborne, and space-based instruments. Many informative lines are detectable currently; nearly all of the important lines should soon be detected by the Infrared Space Observatory.