She was 'burning for' a chance to prove herself. [=she wanted very much to get a chance to prove herself]
('humorous') “We were talking about you last night.” “That explains why I 'felt my ears burning'.”
She was 'burning with' anger at his rudeness. [=she was very angry because of his rudeness]
(..) His spirit will find its way to the halls of your fathers. They had no warning. They were unarmed. Now the Wild Men are moving through the Westfold, burning as they go. Rick, cot and tree. Where is Mama? This is but a taste of the terror that Saruman will unleash. (..)2002 The Lord of the Rings: The Two Towers
(..) Never heard of a Hobbit before. Sounds like Orc mischief to me! They come with fire. They come with axes. Gnawing, biting, breaking, hacking, burning! - Destroyers and usurpers! Curse them! - No! You don't understand. We're Hobbits! Halflings! Shire-folk! (..)2002 The Lord of the Rings: The Two Towers