His money burns a hole in his pocket. He cannot keep it in his pocket, or forbear spending it.
“He has been bolstering up these rotten iron-works. I told him he would burn his fingers.” —Mrs. Lynn Linton.
You cannot burn the candle at both ends. You cannot do two opposite things at one and the same time; you cannot exhaust your energies in one direction, and yet reserve them unimpaired for something else. If you go to bed late you cannot get up early. You cannot eat your cake and have it too. You cannot serve God and Mammon. You cannot serve two masters. Poursuis deux lièvres, et les manques. (La Fontaine.) Simul sorbere ac flare non possum.
We burn daylight. We waste time in talk instead of action. (Shakespeare: Merry Wives of Windsor, ii. 1.)
a stream. A variant of bourn (Anglo-Saxon, burne, a brook, as in Winterbourne, Burnham, Swinburn, etc.).
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894
More on Burn from Infoplease: