Personality is generally theorised to be a stable individual difference variable. Research has shown it to be highly stable over time. Thus, from a theoretical perspective it typically has a primacy in causal models.
Stress can be an ambiguous construct. It can refer to the objective existence of stressful stimuli or the way that individuals perceive those stresses.
From a theoretical perspective, personality is related to both the objective experience of stressors and the subjective perception of stressors. In particular, neuroticism refers to dispositions to experience negative emotions. Personality influences the world. For example, people put themselves into situations that may elicit more stressful experiences. Likewise, personality can influence the experience of stress through various mental models, interpretive schemas, rumination tendencies, and so on.
While one can imagine in the extreme that stress can alter personality (e.g., PTSD), in general personality makes more sense as a factor that has a causal influence on both the objective experience and perception of stressful experiences. In this sense, it personality does not mediate the effect of stress on burnout. This theoretically based conclusion is important. Statistical tests of mediation will provide rubbish information if you just plug in three intercorrelated variables like stress, personality, and burnout.
From a theoretical perspective personality should causally influence the objective presence of stress, the subjective perception of stress, and whether someone experiences burnout. It also makes sense that certain personality traits such as neuroticism would causally increase the relationship between objective stressors and subjective stress and ultimately more extreme forms of subjective stress such as burnout. However, in order to really test this idea, you need to get very good measures of objective stressors. Subjective measures of stress are likely to be influenced directly by personality which is likely to reduce the potential moderator relationship. Alternatively, you might just find that main effects are enough; i.e., that personality and objective stressors mutually contribute to burnout, but that there is no interaction effect (as would be implied by the moderator hypothesis).
There are thousands of empirical articles looking at correlates of personality, stress, strain, burnout and so on. I have read a lot of them many years ago. This answer focuses more on the theoretical underpinnings of any relationship.