Rosemary (Rosmarinus officinalis [family Lamiaceae]), an herb of economic and gustatory repute, is employed in traditional medicines in many countries. Rosemary contains carnosic acid (CA) and carnosol (CS), abietane-type phenolic diterpenes, which account for most of its biological and pharmacological actions, although claims have also been made for contributions of another constituent, rosmarinic acid. This review focuses on the potential applications of CA and CS for Alzheimer’s disease (AD), Parkinson’s disease (PD), and coronavirus disease 2019 (COVID-19), in part via inhibition of the NLRP3 inflammasome. CA exerts antioxidant, anti-inflammatory, and neuroprotective effects via phase 2 enzyme induction initiated by activation of the KEAP1/NRF2 transcriptional pathway, which in turn attenuates NLRP3 activation. In addition, we propose that CA-related compounds may serve as therapeutics against the brain-related after-effects of SARSCoV-2 infection, termed “long-COVID.” One factor that contributes to COVID-19 is cytokine storm emanating from macrophages as a result of unregulated inflammation in and around lung epithelial and endovascular cells. Additionally, neurological aftereffects such as anxiety and “brain fog” are becoming a major issue for both the pandemic and post-pandemic period. Many reports hold that unregulated NLRP3 inflammasome activation may potentially contribute to the severity of COVID-19 and its aftermath. It is therefore possible that suppression of NLRP3 inflammasome activity may prove efficacious against both acute lung disease and chronic neurological after-effects. Because CA has been shown to not only act systemically but also to penetrate the blood–brain barrier and reach the brain parenchyma to exert neuroprotective effects, we discuss the evidence that CA or rosemary extracts containing CA may represent an effective countermeasure against both acute and chronic pathological events initiated by SARS-CoV-2 infection as well as other chronic neurodegenerative diseases including AD and PD.