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]
(..) Maggie, it's just a tick. - Get a match. - We're not going to burn him, Jordan. It'll have to back out. Can't leave the head in. - We can't just be burning tick heads. JORDAN: Then get some alcohol. I don't have any. JORDAN: You don't have any alcohol? MAGGIE: I don't operate here. JORDAN: (..)1998 City of Angels