Influenza A viruses cause a mild-to-severe respiratory disease that affects millions of people each year. One of the many determinants of disease outcome is the innate immune response to the viral infection. While antiviral responses are essential for viral clearance, excessive innate immune activation promotes lung damage and disease. The influenza A virus RNA polymerase is one of viral proteins that affect innate immune activation during infection, but the mechanisms behind this activity are not well understood. In this review, we discuss how the viral RNA polymerase can both activate and suppress innate immune responses by either producing immunostimulatory RNA species or directly targeting the components of the innate immune signalling pathway, respectively. Furthermore, we provide a comprehensive overview of the polymerase residues, and their mutations, associated with changes in innate immune activation, and discuss their putative effects on polymerase function based on recent advances in our understanding of the influenza A virus RNA polymerase structure.