As big data and artificial intelligence (AI) technologies become embedded in the governance of higher education, algorithmic decision-making has improved managerial efficiency but has also led teachers to question the fairness of evaluation and to face difficulties in psychological adaptation, making job burnout an increasingly salient concern. How perceived fairness of algorithmic decision-making (FAD) relates to university teachers’ job burnout, and through what psychological mechanism, remains unclear. Drawing on the transactional theory of stress and organizational support theory, we built a moderated chain-mediation model in which psychological safety and algorithm anxiety operate as sequential mediators and organizational support acts as a moderator. Questionnaire data were collected from 895 university teachers in Shanghai and Gansu, China, and the model was tested with the SPSS PROCESS macro using standardized coefficients and bias-corrected bootstrapping. Three findings emerged. First, FAD was negatively associated with job burnout, and this relationship was fully accounted for by the proposed mediators. Second, psychological safety and algorithm anxiety each mediated the link and together formed a significant serial path (FAD → psychological safety → algorithm anxiety → job burnout; β = −0.002, 95% CI [−0.005, −0.000]); algorithm anxiety was the single strongest mediator, accounting for 23.60% of the total effect. Third, organizational support moderated the algorithm anxiety–burnout association: Johnson–Neyman analysis indicated a significant positive association at low support (below 4.16) that became non-significant in the mid-range and, above 5.36, reversed in sign. Because the design is cross-sectional, this reversal should be read as a boundary pattern that requires replication rather than as evidence of a causal switch. By situating FAD, psychological safety, algorithm anxiety, and job burnout within one framework, the study clarifies the technology–individual–organization interplay and offers evidence for fairer algorithmic governance and for protecting teachers’ occupational health.