The global burden of coronavirus disease 2019 (COVID-19) to public health and global economy has stressed the need for rapid and simple diagnostic methods. From this perspective, plasmonic-based biosensing can manage the threat of infectious diseases by providing timely virus monitoring. In recent years, many plasmonics’ platforms have embraced the challenge of offering on-site strategies to complement traditional diagnostic methods relying on the polymerase chain reaction (PCR) and enzyme-linked immunosorbent assays (ELISA). This review compiled recent progress on the development of novel plasmonic sensing schemes for the effective control of virus-related diseases. A special focus was set on the utilization of plasmonic nanostructures in combination with other detection formats involving colorimetric, fluorescence, luminescence, or Raman scattering enhancement. The quantification of different viruses (e.g., hepatitis virus, influenza virus, norovirus, dengue virus, Ebola virus, Zika virus) with particular attention to Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) was reviewed from the perspective of the biomarker and the biological receptor immobilized on the sensor chip. Technological limitations including selectivity, stability, and monitoring in biological matrices were also reviewed for different plasmonic-sensing approaches.