In the solar corona, the free energy, i.e., the excess in magnetic energy over a ground-state potential field, forms the reservoir of energy that can be released during solar flares and coronal mass ejections. Such free energy provides a measure of the magnetic field nonpotentiality. Recent theoretical and observational studies indicate that the presence of nonpotential magnetic fields is imprinted into the structures of infrared, off-limb, coronal polarization. In this paper, we investigate the possibility of exploiting such observations for mapping and studying the accumulation and release of coronal free magnetic energy, with the goal of developing a new tool for identifying "hot spots" of coronal free energy such as those associated with twisted and/or sheared coronal magnetic fields. We applied forward modeling of infrared coronal polarimetry to three-dimensional models of nonpotential and potential magnetic fields. From these we defined a quantitative diagnostic of nonpotentiality that in the future could be calculated from a comparison of infrared, off-limb, coronal polarization observations from, e.g., the Coronal Multi-channel Polarimeter or the Daniel K. Inouye Solar Telescope, and the corresponding polarization signal forward-modeled from a potential field extrapolated from photospheric magnetograms. We considered the relative diagnostic potential of linear and circular polarization, and the sensitivities of these diagnostics to coronal density distributions and assumed boundary conditions of the potential field. Our work confirms the capacity of polarization measurements for diagnosing nonpotentiality and free energy in the solar corona.