Highly infectious viral diseases are a serious threat to mankind as they can spread rapidly among the community, possibly even leading to the loss of many lives. Early diagnosis of a viral disease not only increases the chance of quick recovery, but also helps prevent the spread of infections. There is thus an urgent need for accurate, ultrasensitive, rapid, and affordable diagnostic techniques to test large volumes of the population to track and thereby control the spread of viral diseases, as evidenced during the COVID-19 and other viral pandemics. This review paper critically and comprehensively reviews various emerging nanophotonic biosensor mechanisms and biosensor technologies for virus detection, with a particular focus on detection of the SARS-CoV-2 (COVID-19) virus. The photonic biosensing mechanisms and technologies that we have focused on include: (a) plasmonic field enhancement via localized surface plasmon resonances, (b) surface enhanced Raman scattering, (c) nano-Fourier transform infrared (nano-FTIR) near-field spectroscopy, (d) fiber Bragg gratings, and (e) microresonators (whispering gallery modes), with a particular emphasis on the emerging impact of nanomaterials and two-dimensional materials in these photonic sensing technologies. This review also discusses several quantitative issues related to optical sensing with these biosensing and transduction techniques, notably quantitative factors that affect the limit of detection (LoD), sensitivity, specificity, and response times of the above optical biosensing diagnostic technologies for virus detection. We also review and analyze future prospects of cost-effective, lab-on-a-chip virus sensing solutions that promise ultrahigh sensitivities, rapid detection speeds, and mass manufacturability.