The US Census Bureau will implement a new privacy-preserving disclosure avoidance system (DAS), which includes application of differential privacy, on the public-release 2020 census data. There are concerns that the DAS may bias small-area and demographically-stratified population counts, which play a critical role in public health research and policy, serving as denominators in estimation of disease/mortality rates. Employing three DAS demonstration products, we quantify errors attributable to reliance on DAS-protected denominators in standard small-area disease mapping models for characterizing health inequities. We conduct simulation studies and real data analyses of inequities in premature mortality at the census tract level in Massachusetts. Results show that overall patterns of inequity by racialized group and economic deprivation level are not compromised by the DAS. While early versions of DAS induce errors in mortality rate estimation that are larger for Black than for non-Hispanic white populations, this issue is ameliorated in newer DAS versions.