COVID-19 has led to an unprecedented strain on healthcare workers (HCWs). This study aimed to determine the prevalence of burnout in hospital employees during a prolonged pandemic-induced burden on healthcare systems. An online survey among employees of a Czech and Slovak university hospital was conducted between November 2021 and January 2022, approximately when the incidence rates peaked in both countries. The Maslach Burnout Inventory-Human Services Survey was applied. We obtained 807 completed questionnaires (75.1% from Czech employees, 91.2% from HCWs, 76.2% from women; mean age of 42.1 ± 11 years). Burnout in emotional exhaustion (EE) was found in 53.2%, depersonalization (DP) in 33%, and personal accomplishment (PA) in 47.8% of respondents. In total, 148 (18.3%) participants showed burnout in all dimensions, 184 (22.8%) in two, and 269 (33.3%) in at least one dimension. Burnout in EE and DP (65% and 43.7%) prevailed in physicians compared to other HCWs (48.6% and 28.8%). Respondents from COVID-19-dedicated units achieved burnout in the EE and DP dimensions with higher rates than non-frontline HCWs (58.1% and 40.9% vs. 49.9% and 27.7%). Almost two years of the previous overloading of healthcare services, caused by the COVID-19 pandemic, resulted in the relatively high prevalence of burnout in HCWs, especially in physicians and frontline HCWs.