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 a free energy further provides a measure of non-potentiality of the magnetic field. Recent theoretical studies indicate that the presence of non-potential magnetic fields is imprinted into the structures of infrared, off-limb, coronal polarization. Thanks to the Coronal Multichannel Polarimeter, coronal linear polarization observations are now available on a daily basis and have been demonstrated to be useful for diagnosing properties of the coronal magnetic field. Future telescopes such as the Daniel K. Inouye Solar Telescope and the proposed Coronal Solar Magnetism Observatory Large Coronagraph will yield circular polarization measurements in the corona. 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 non-potential and potential magnetic fields. From these we defined a diagnostic of non-potentiality that in future could be calculated from a comparison of infrared, o-limb, coronal polarization observations 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 non-potentiality and free energy in the solar corona.