SARS-CoV-2 is a novel coronavirus that infects cells through the angiotensin-converting enzyme 2 receptor, aided by proteases that prime the spike protein of the virus to enhance cellular entry. Neuropilin 1 and 2 (NRP1 and NRP2) act as additional viral entry factors. SARS-CoV-2 infection causes COVID-19 disease. There is now strong evidence for neurological impacts of COVID-19, with pain as an important symptom, both in the acute phase of the disease and at later stages that are colloquially referred to as “long COVID.” In this narrative review, we discuss how COVID-19 may interact with the peripheral nervous system to cause pain in the early and late stages of the disease. We begin with a review of the state of the science on how viruses cause pain through direct and indirect interactions with nociceptors. We then cover what we currently know about how the unique cytokine profiles of moderate and severe COVID-19 may drive plasticity in nociceptors to promote pain and worsen existing pain states. Finally, we review evidence for direct infection of nociceptors by SARS-CoV-2 and the implications of this potential neurotropism. The state of the science points to multiple potential mechanisms through which COVID-19 could induce changes in nociceptor excitability that would be expected to promote pain, induce neuropathies, and worsen existing pain states.