Tut, dun's the mouse, the constable's own word.
If thou art dun, we'll draw thee from the mire
Of this - save your reverence - love, wherein thou stick'st
Up to the ears. Come, we burn daylight, ho!
--Mercutio, Romeo and Juliet I.IV
Burning daylight is an old English idiom, meaning that one is wasting time -- particularly waiting time when doing a task that requires daylight, and which therefor must be finished before the sun sets. This is apparently a literalization of the original meaning, for when Romeo himself takes it literally, claiming that they could not be burning daylight as it is nighttime, Mercutio scolds him for his foolishness, and goes on to explain: "I mean, sir, in delay / We waste our lights in vain, like lamps by day." Clear evidence that modern folk (and romantic fools) can't handle the abstract level of language that our ancestors glorified in.
We don't know for certain if Shakespeare coined the phrase himself, but he is often credited with doing so. He used the phrase a second time, in The Merry Wives of Windsor1, 2. Wherever it first appeared, it has proven surprisingly popular, and in addition to being in common usage 400 years later, it has also been used as a name for various movies, songs, bands, books3, and businesses.
1. Someone please node The Merry Wives of Windsor already.
2. Not an interesting quote, I'm afraid: Mrs. Ford: "We burn daylight: here, read, read; perceive how I might be knighted."
3. It may be worth noting that the most famous of these books, Jack London's Burning Daylight (1910), uses it as an Indian name, not in the sense of the popular idiom.