The severe acute respiratory syndrome-related coronavirus 2 (SARS-CoV-2) and the сoronavirus disease 2019 (COVID-19) have become a global health threat. At the height of the pandemic, major efforts were focused on reducing COVID-19-associated morbidity and mortality. Now is the time to study the long-term effects of the pandemic, particularly cognitive impairment associated with long COVID. In recent years much attention has been paid to the possible relationship between COVID-19 and Alzheimer’s disease, which is considered a main cause of age-related cognitive impairment. Genetic predisposition was shown for both COVID-19 and Alzheimer’s disease. However, the analysis of the similarity of the genetic architecture of these diseases is usually limited to indicating a positive genetic correlation between them. In this review, we have described intrinsic linkages between COVID-19 and Alzheimer’s disease, pointed out shared susceptibility genes that were previously identified in genome-wide association studies of both COVID-19 and Alzheimer’s disease, and highlighted a panel of SNPs that includes candidate genetic risk markers of the long COVID-associated cognitive impairment.