Background: Headaches are a common symptom in healthcare workers (HCWs), mainly associated with high levels of stress. Different research has studied their incidence during the COVID-19 pandemic, most of them with correlational designs, and at the beginning of the pandemic and focused on the associated occupational variables. Aims: (1) To analyze the incidence of headaches in HCWs at the beginning of the COVID-19 pandemic and their maintenance six months later. (2) To explore the risk factors associated with their onset and maintenance, including sociodemographic, occupational, emotional symptomatology, and personality variables. (3) To propose a model to explain the chronification of stress in burnout, including the moderating role of chronic headaches. Methods: A prospective study (n = 259 HCWs) at three points in time during the COVID-19 pandemic, from the alarm state phase (T1: May–June 2020) to the post-pandemic stage (T3: April–July 2022), including an intermediate measure six months after T1 (T2). Descriptive analyses, Pearson’s chi-square, Student’s t, logistic regressions, and moderated mediation models were conducted using the Process package for SPSS. In addition to headaches, socio-demographic, occupational, emotional symptomatology, and personality variables were included. Results: At T1 the prevalence of headaches was 69.9%. At T2 the prevalence was 73.7%. Of these, 59.5% are T1–T2 sustained headaches. Headaches at T1 were associated with age (p = 0.010) (younger HCWs), professional category (p = 0.049) (nurses), service (p = 0.023) (ICU, COVID hospitalization), non-availability of PPE (p = 0.010), additional COVID-19 symptomatology (p < 0.001), and concern for contagion of family members (p < 0.001) (higher scores). In addition, HCWs with headaches had higher levels of stress (p = 0.001), anxiety (p = 0.001), depression (p = 0.041), and sleep disorders (p < 0.001). A subsequent logistic regression analysis showed that of the above variables, the presence of additional COVID-19 symptoms (p < 0.001) and depression (p = 0.010) were the predictor variables. With regard to the maintenance of headaches (T1–T2), anxiety (p = 0.035), stress (p = 0.001), and cognitive fusion (p = 0.013) were found to be the significant variables. The tested model proposes anxiety (T1) as antecedent, cognitive fusion (T2) as mediator, burnout (T3) as consequent, and chronic headaches (yes/no) as the moderating variable between anxiety and burnout (model 5). The model is significant (F = 19.84, p < 0.001) and contributes to the explanation of 36% of the variance of burnout. The relationships in the model are all statistically significant, and specifically chronic headaches contribute to a 6-fold increase in the likelihood of burnout. Conclusions: The present research differentiates between precipitating and maintenance factors of headaches in HCWs. The former, more studied in previous research, are usually related to sociodemographic and occupational variables and levels of anxiety and stress. Maintenance factors, scarcely explored, are related to the maintenance of emotional symptomatology and the inability to manage intrusive thoughts (i.e., cognitive fusion). Of particular interest is that the presence of chronic headaches itself is capable of producing burnout as a post-COVID syndrome.