Cellular senescence is a state of stable cell cycle arrest associated with macromolecular alterations and secretion of pro-inflammatory cytokines and molecules. Senescence-associated phenotypes restrict damage propagation and activate immune responses, two essential processes involved in response to viral infections. However, excessive accumulation and persistence of senescent cells can become detrimental and promote pathology and dysfunctions. Various pharmacological interventions, including antiviral therapies, lead to aberrant and premature senescence. Here, we review the molecular mechanisms by which viral infections and antiviral therapy induce senescence. We highlight the importance of these processes in attenuating viral dissemination and damage propagation, but also how prematurely induced senescent cells can promote detrimental adverse effects in humans. We describe which sequelae due to viral infections and treatment can be partly due to excessive and aberrant senescence. Finally, we propose that pharmacological strategies which eliminate senescent cells or suppress their secretory phenotype could mitigate side effects and alleviate the onset of additional morbidities. These strategies can become extremely beneficial in patients recovering from viral infections or undergoing antiviral therapy.