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]
(..) There's not much time left in the world, Sarah. Kyle, don't go! Kyle! It's like a giant strobe light, burning right to my eyes. Somehow l can still see. Look, we know the dream's the same every night. Why do we have to... Please continue. (..)1991 Terminator 2: Judgment Day