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 was nothing l could do. -He just caught fire. -And about time too. He's been looking dreadful for days. Pity you had to see him on a burning day. Fawkes is a phoenix, Harry. They burst into flame when it is time for them to die. . . . . .and then they are. . . . . .reborn from the ashes. (..)2002 Harry Potter and the Chamber of Secrets