The existence of a charge density wave (CDW) in transition-metal dichalcogenide (TMDC) CuS₂ has remained undetermined since its first experimental synthesis nearly 50 years ago. Despite conflicting experimental literature regarding its low-temperature structure, there exists no theoretical investigation of the phonon properties and lattice stability of this material. By studying the first-principles electronic structure and phonon properties of CuS₂ at various electronic temperatures, we identify temperature-sensitive soft phonon modes which unveil a previously unreported Kohn anomaly at approximately 100 K. Variation of the electronic temperature shows the presence of two distinct phases, characterized at low electronic temperature by a 2×2×2 periodic charge modulation associated with the motion of the S₂ dimers. We find this is driven by a slight orbital occupation imbalance of the copper d and sulfur p orbitals, reminiscent of the Jahn-Teller effect in finite systems. Investigation of the Fermi surface presents a potential Fermi surface nesting vector related to the location of the Kohn anomaly and observed band splittings in the unfolded band structure. The combination of these results suggests a strong possibility of CDW order in CuS₂. Further study of CuS₂ in monolayer form finds no evidence of a CDW phase, as the identified bulk periodic distortions cannot be realized in two dimensions. This behavior sets this material apart from other transition-metal dichalcogenide materials, which exhibit a charge density wave phase down to the two-dimensional limit. As CDW in TMDC materials is considered to compete with superconductivity, the lack of a CDW in monolayer CuS₂ suggests the possibility of enhanced superconductivity relative to other transition-metal dichalcogenides. Overall, our work identifies CuS₂ as a previously unrealized candidate to study the interplay of superconductivity, CDW order, and dimensionality.