Getting admission into a university is highly competitive globally and can be regarded as a mental stressor for some individuals. In Bangladesh, most of the universities allow students to repeat the test, whereas repeat test-taking students are supposed to suffer from psychological issues like burnout because of academic non-achievement in their first attempt. But there is no prior study assessing burnout to the population, which was investigated herein. A cross-sectional study was carried out among a total of 911 Bangladeshi university entrance test-taking students (i.e., 49.5% first-time test-takers). The survey measures included socio-demographics, admission-related issues, and mental health problems (depression, anxiety, and burnout). Of the mental health problems, depression, anxiety, and burnout were assessed by the Patient Health Questionnaire (PHQ-9), Generalized anxiety disorder (GAD-7), and Maslach Burnout Inventory—Student Survey (MBI–SS), respectively. In light of two-dimensional and three-dimensional cutoff schemes of the MBI—SS, about 43.7% and 17.3% of the test-taking students, respectively, were classified to be burnout in the total sample. Additionally, 48.1%, 71.4%, and 49.1% of the students in the total sample reported higher emotional exhaustion, higher cynicism, and lower academic efficacy, respectively. In respect to the student status, two-dimensional burnout (48.0% vs. 39.2%; χ2 = 7.164, p = 0.007), and emotional exhaustion (52.2% vs. 43.9%, χ2 = 6.789, p = 0.034) were significantly higher among the repeat test-taking students. Satisfaction with previous mock tests, depression, and anxiety were the associated factors of burnout in all three samples. Based on the present study, it can be concluded that the university entrance test appearing students are highly prevalent to burnout, and the rate increases with academic non-achievement. As the first study assessing burnout to the population, the findings are anticipated to be helpful for policymaking and further studies both inside and outside Bangladesh.