。
She was burning with jealousy.
Literal
She [topic-は] jealousy [with-に] was-burning.
嫉妬に燃える casts jealousy as a fire — she herself was burning, consumed by the emotion. 嫉妬に marks the jealousy as the cause or fuel of the burning, with the intransitive 燃える (to burn) describing her state rather than what she does. The ~ていた places this in the past as a sustained ongoing condition — she was in the grip of jealousy for a stretch of time, not just a flash. The fire-as-jealousy metaphor cuts across many languages, but Japanese reaches for it especially often with negative emotions like 怒り (anger) and 恨み (resentment) too.