I've been working with Agile for years and I worked with people who burned out, but there was not even a single case where Agile contributed to burning out, directly or indirectly. In fact, Agile contributed to unload pressure off developers and prevent people from overworking and burning out.

The main factors in burning out we're always time ranges from the enforcement of unrealistic schedules and poor managerial/team culture. It's not Agile's fault that your manager wants a feature out in half the time while looming dismissals over your head.

It's not Agile's fault that stack ranking developers results in hostile team environments where team members don't help out people and even go as far as putting roadblocks elsewhere so that they aren't the ones in the critical path. Agile explicitly provides the tools to make each one of these burnout-inducing scenarios as non-issues.