The appearance and quick spread of the new severe acute respiratory syndrome coronavirus disease, COVID-19, brought major societal challenges. Importantly, suitable medical diagnosis procedures and smooth clinical management of the disease are an emergent need, which must be anchored on novel diagnostic methods and devices. Novel molecular diagnostic tools relying on nucleic acid amplification testing have emerged globally and are the current gold standard in COVID-19 diagnosis. However, the need for widespread testing methodologies for fast, effective testing in multiple epidemiological scenarios remains a crucial step in the fight against the COVID-19 pandemic. Biosensors have previously shown the potential for cost-effective and accessible diagnostics, finding applications in settings where conventional, laboratorial techniques may not be readily employed. Paper- and cellulose-based biosensors can be particularly relevant in pandemic times, for the renewability, possibility of mass production with sustainable methodologies, and safe environmental disposal. In this review, paper-based devices and platforms targeting SARS-CoV-2 are showcased and discussed, as a means to achieve quick and low-cost PoC diagnosis, including detection methodologies for viral genomic material, viral antigen detection, and serological antibody testing. Devices targeting inflammatory markers relevant for COVID-19 are also discussed, as fast, reliable bedside diagnostic tools for patient treatment and follow-up.