Dengue virus (DENV) cycles between mosquito and mammalian hosts. To examine how DENV populations adapt to these different host environments, we used serial passage in human and mosquito cell lines and estimated fitness effects for all single-nucleotide variants in these populations using ultra-deep sequencing. This allowed us to determine the contributions of beneficial and deleterious mutations to the collective fitness of the population. Our analysis revealed that the continuous influx of a large burden of deleterious mutations counterbalances the effect of rare, host-specific beneficial mutations to shape the path of adaptation. Beneficial mutations preferentially map to intrinsically disordered domains in the viral proteome and cluster to defined regions in the genome. These phenotypically redundant adaptive alleles may facilitate host-specific DENV adaptation. Importantly, the evolutionary constraints described in our simple system mirror trends observed across DENV and Zika strains, indicating it recapitulates key biophysical and biological constraints shaping long-term viral evolution.