The resurgence of influenza and continued circulation of SARS-CoV-2 raise the question of how these viruses interact in a co-exposed host. Here we studied virus-virus and host-virus interactions during influenza A virus (IAV) -SARS-CoV-2 coinfection using differentiated cultures of the human airway epithelium. Coexposure to IAV enhanced the tissue antiviral response during SARS-CoV-2 infection and suppressed SARS-CoV-2 replication. Oseltamivir, an antiviral targeting influenza, reduced IAV replication during coinfection but also reduced the antiviral response and paradoxically restored SARS-CoV-2 replication. These results highlight the importance of diagnosing coinfections and compel further study of how coinfections impact the outcome of antiviral therapy.