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Many biological processes discriminate between correct and incorrect substrates through the kinetic proofreading mechanism that enables lower error at the cost of higher energy dissipation. Elucidating physico-chemical constraints for global minimization of dissipation and error is important for understanding enzyme evolution. Here, we identify theoretically a fundamental error–cost bound that tightly constrains the performance of proofreading networks under any parameter variations preserving the rate discrimination between substrates. The bound is kinetically controlled, i.e. completely determined by the difference between the transition state energies on the underlying free energy landscape. The importance of the bound is analysed for three biological processes. DNA replication by T7 DNA polymerase is shown to be nearly optimized, i.e. its kinetic parameters place it in the immediate proximity of the error–cost bound. The isoleucyl-tRNA synthetase (IleRS) of E. coli also operates close to the bound, but further optimization is prevented by the need for reaction speed. In contrast, E. coli ribosome operates in a high-dissipation regime, potentially in order to speed up protein production. Together, these findings establish a fundamental error–dissipation relation in biological proofreading networks and provide a theoretical framework for studying error–dissipation trade-off in other systems with biological discrimination.