Emotional exhaustion, cynicism, and work inefficiency are three dimensions that define burnout syndrome among teachers. On another note, resilience can be understood as the ability to adapt to the environment and overcome adverse situations. In addition, COVID-19 has provided a threatening environment that has led to the implementation of resilience strategies to struggle with burnout and cope with the virus. The aim of this study was to analyze the relationship between resilience, burnout dimensions, and variables associated with COVID-19 through the design of an artificial neural network architecture. For this purpose, the Maslach Burnout Inventory-General Survey (MBI-GS), the Brief Resilience Coping Scale (BRCS), and a questionnaire on stress towards COVID-19 were administered to 419 teachers from secondary schools in southeastern Spain (292 females; 69.7%). The results showed that 30.8% suffered from burnout (high emotional exhaustion, high cynicism, and low professional efficacy) and that 38.7% had a high level of resilience, with an inverse relationship between both constructs. Likewise, we modelled an ANN able to predict burnout syndrome among 97.4% of teachers based on its dimensions, resilience, sociodemographic variables, and the stress generated by COVID-19. Our conclusions shed some light on the efficacy of relying on artificial intelligence in the educational field to predict the psychological situation of teachers and take early action.