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]
(..) But you heard all of this before, haven't you? On the Long Bridge of Volantis. The dragons will purify nonbelievers by the thousands, burning their sins and flesh away. Ideally we'd avoid purifying too many nonbelievers. The Mother of Dragons has followers of many different faiths. (..)2016 TvShow: Game of Thrones Title: The Door Season: 6 Episode 5