Viruses invade a host through infecting and spreading among host cells. Initial virus replication and transmission are counteracted by the host innate immune response, in particular the interferon response. Although the virus-innate immune interaction has been studied in laboratory for a long time, a theoretical understanding of how the interferon response impacts on viral spread is lacking. In this work, we model this interaction as a competition process between the virus spreading and the interferon response on a two-layer multiplex network with virus and interferon spread on the two layers separately. We specifically explore how the overlap between the two layers impacts on the threshold and the final size of virus spread. A mean-field method and a general homogeneous multiplex network are adopted to approximate and analyze the behavior of system. We find that interferon response can effectively stop the spread of the virus or reduce the final size of viral infection when the two networks largely overlap each other. This is true especially when the interferon response is strong. The results provide insights about how the innate immune response counteracts viral invasion and spread. It may also have implications for designing strategies for risk mitigation in computer or social networks.