Estimation of natural selection on protein-coding sequences is a key comparative genomics approach for de novo prediction of lineage specific adaptations. Selective pressure is measured on a per-gene basis by comparing the rate of non-synonymous substitutions to the rate of neutral evolution, typically assumed to be the rate of synonymous substitutions. All published codon substitution models have been time-reversible and thus assume that sequence composition does not change over time. We previously demonstrated  that if time-reversible DNA substitution models are applied blindly in the presence of changing sequence composition, the number of substitutions is systematically biased towards overestimation. We extend these findings to the case of codon substitution models and further demonstrate that the ratio of non-synonymous to synonymous rates of substitutions tends to be underestimated over three data sets of insects, mammals, and vertebrates. Our basis for comparison is a non-stationary codon substitution model that is heterogeneous across lineages and allows sequence composition to change. Model selection and model fit results demonstrate that our new model tends to fit the data better. Direct measurement of non-stationarity shows that bias in estimates of natural selection and genetic distance increases with the violation of the stationarity assumption. Additionally, inferences drawn under time-reversible models are systematically affected by compositional divergence. As genomic sequences accumulate at an accelerating rate, the importance of accurate de novo estimation of natural selection increases. Our results establish that our new model provides a more robust perspective on this fundamental quantity.