Generative AI (GenAI) is rapidly reshaping software development workflows. While prior studies emphasize productivity gains, the adoption of GenAI also introduces new pressures that may harm developers' well-being. In this paper, we investigate the relationship between the adoption of GenAI and developers' burnout. We utilized the Job Demands--Resources (JD--R) model as the analytic lens in our empirical study. We employed a concurrent embedded mixed-methods research design, integrating quantitative and qualitative evidence. We first surveyed 442 developers across diverse organizations, roles, and levels of experience. We then employed Partial Least Squares--Structural Equation Modeling (PLS-SEM) and regression to model the relationships among job demands, job resources, and burnout, complemented by a qualitative analysis of open-ended responses to contextualize the quantitative findings. Our results show that GenAI adoption heightens burnout by increasing job demands, while job resources and positive perceptions of GenAI mitigate these effects, reframing adoption as an opportunity.