To study the chemical evolution across cosmic epochs, we investigate Ne, S, Cl, and Ar abundance patterns in the Cosmic Origins Spectrograph Legacy Archive Spectroscopic SurveY (CLASSY). CLASSY comprises local star-forming galaxies (SFGs; 0.02 < z < 0.18) with enhanced star formation rates, making them strong analogues to high-z SFGs. With direct measurements of electron temperature, we derive accurate ionic abundances for all elements and assess ionization correction factors (ICFs) to account for unseen ions and derive total abundances. We find Ne/O, S/O, Cl/O, and Ar/O exhibit constant trends with gas-phase metallicity for 12+log(O/H) < 8.5 but significant correlation for Ne/O and Ar/O with metallicity for 12+log(O/H) > 8.5, likely due to ICFs. Thus, the applicability of the ICFs to integrated spectra of galaxies could bias results, underestimating true abundance ratios. Using CLASSY as a local reference, we assess the evolution of Ne/O, S/O, and Ar/O in galaxies at z > 3, finding no cosmic evolution of Ne/O, while the lack of direct abundance determinations for S/O and Ar/O can bias the interpretation of the evolution of these elements. We determine the fundamental metallicity relationship (FMR) for CLASSY and compare to the high-redshift FMR, finding no evolution. Finally, we perform the first mass–neon relationship analysis across cosmic epochs, finding a slight evolution to high Ne at later epochs. The robust abundance patterns of CLASSY galaxies and their broad range of physical properties provide essential benchmarks for interpreting the chemical enrichment of the early galaxies observed with the JWST.