COVID-19 is an infectious disease caused by the SARS-CoV-2 coronavirus and characterized by an extremely variable disease course, ranging from asymptomatic cases to severe illness. Although all individuals may be infected by SARS-CoV-2, some people, including those of older age and/or with certain health conditions, including cardiovascular disease, diabetes, cancer, and chronic respiratory disease, are at higher risk of getting seriously ill. For cancer patients, there are both direct consequences of the COVID-19 pandemic, including that they are more likely to be infected by SARS-CoV-2 and more prone to develop severe complications, as well as indirect effects, such as delayed cancer diagnosis or treatment and deferred tests. Accumulating data suggest that aberrant SARS-CoV-2 immune response can be attributed to impaired interferon signaling, hyper-inflammation, and delayed adaptive immune responses. Interestingly, the SARS-CoV-2-induced immunological abnormalities, DNA damage induction, generation of micronuclei, and the virus-induced telomere shortening can abnormally activate the DNA damage response (DDR) network that plays a critical role in genome diversity and stability. We present a review of the current literature regarding the molecular mechanisms that are implicated in the abnormal interplay of the immune system and the DDR network, possibly contributing to some of the COVID-19 complications.