According to the Makishema–Mackenzie model assumption, the dissociation energy and packing density for a quaternary TeO2-As2O3-B2O3-Li2O glass system were evaluated. The dissociation energy rose from 67.07 to 71.85 kJ/cm3, whereas the packing factor decreased from 16.55 to 15.21 cm3/mol associated with the replacement of TeO2 by LiO2 compounds. Thus, as a result, the elastic moduli (longitudinal, shear, Young, and bulk) were enhanced by increasing the LiO2 insertion. Based on the estimated elastic moduli, mechanical properties such as the Poisson ratio, microhardness, longitudinal velocity, shear velocity, and softening temperature were evaluated for the investigated glass samples. In order to evaluate the studied glasses’ gamma-ray shield capacity, the MCNP-5 code, as well as a theoretical Phy-X/PSD program, were applied. The best shielding capacity was achieved for the glass system containing 25 mol% of TeO2, while the lowest ability was obtained for the glass sample with a TeO2 concentration of 5 mol%. Furthermore, a correlation between the studied glasses’ microhardness and linear attenuation coefficient was performed versus the LiO2 concentration to select the glass sample which possesses a suitable mechanical and shielding capacity.