In this paper we report a detailed analysis of the temperature-dependent optical properties of epitaxially grown cadmium arsenide (Cd$_3$As$_2$), a newly discovered three-dimensional Dirac semimetal. Dynamic Fermi level tuning -- instigated from Pauli-blocking in the linear Dirac cone -- and varying Drude response, generate large variations in the mid and far-infrared optical properties. We demonstrate thermo-optic shifts larger than those of traditional III-V semiconductors, which we attribute to the obtained large thermal expansion coefficient as revealed by first-principles calculations. Electron scattering rate, plasma frequency edge, Fermi level shift, optical conductivity, and electron effective mass analysis of Cd$_3$As$_2$ thin-films are quantified and discussed in detail. Our ab initio density functional study and experimental analysis of epitaxially grown Cd$_3$As$_2$ promise applications for nanophotonic and nanoelectronic devices, such as reconfigurable metamaterials and metasurfaces, nanoscale thermal emitters, and on-chip directional antennas.