I can only think of the following words to replace burn out, but 'am sure you know this all ready
agobiado,agotado,harto. I, my self would use agotado.

Porque lo que quema es
porque lo que agota.........

El "burn-out " se produce
el agotamiento se produce
It is an interesting question.

To try to translate the word exactly like in English might be a mistake, because you don't translate "Eso tiene gracia" as "That has 'gracia'", simply because "fun" and "grace" don't fit here, but "That is funny", where the whole idea is expressed with a completely different construction, so it sounds like English.

in Sevilla people say things like "Tengo una quema(d)era de tanto currar esta semana que no veas...", and "Llevo un quemón encima de echar tantas horas en el curro...", but surely they are not standard words recognized elsewhere. To make a more standard translation that sentence would have to be re-phrased like:

Uno se termina quemando si trabaja con personas.

And that solves the problem: natural Spanish again.

Gus, no me parece mala idea: el agotamiento psicológico quizás...

It's used as both a verb and a noun. Examples:

The in-service for administrators will discuss the problem of burn-out among young teachers.
If you don't take care of yourself, you'll burn out before you've been in the ministry for long.

And a saying: I'd rather burn out than fade away.

From the American Heritage Dictionary of the English language:

burn out. 3. To become exhausted, especially as a result of overwork or dissipation.

Gus's and Lazarus' suggestions both sound really good.

Think of a candle or lantern. If you change the metaphor to cars, it becomes run out of gas

And this from Edna St. Vincent Millay:

My candle burns at both ends;
It will not last the night;
But ah, my foes, and oh, my friends?
It gives a lovely light.