Objectives: The social support of caregivers is fundamental, and their quality of life, mental health, and the burden of caring are related to the social support they receive during their interaction with people. Because the coronavirus pandemic and related quarantine have affected people’s participation and social support, the caregivers’ lifestyle has changed, and they have become more isolated and lonely. This study aims to investigate the mediating role of the burden of care in the relationship between social support and mental health of family caregivers of elderly Iranians with chronic diseases of the nervous system during the coronavirus pandemic. Methods: We recruited a sample of 249 family caregivers of the elderly with Alzheimer, Parkinson, and stroke. The study data were collected by electronic questionnaires of perceived social support of Zimet, Novak care burden, and Goldberg mental health questionnaire. The obtained data were analyzed using the Pearson correlation and structural equation modeling. Results: This study showed that family caregivers experience high social support, a moderate care load, and their mental health is on the verge of illness during the coronavirus epidemic. The results of the Pearson correlation and structural equation modeling showed a positive and significant relationship between social support perceived by caregivers and their mental health, and the burden of care has a mediating role in this relationship. Discussion: Finally, because social support is effective in reducing the distress experienced by caregivers and improving their performance and mental health, future plans and interventions are expected to consider strengthening social support as primary prevention to protect caregivers from mental health symptoms.