Burn one's boats/bridges - stake everything on success
Or, more accurately, destroy one's own means of retreat should a venture fail - an occasional practice of some Roman generals to stiffen the resolve of their troops against the possibility of any such failure. Curiously, both expressions are recorded in English no earlier than the last 20 years of the 19th century.
Burn the candle at both ends - exhaust oneself
In the days when candles were a customary form of lighting, burning them at both ends was synonymous with wastefulness. The modern meaning is milder; in some contexts, it implies anxiety for someone's state of health rather than a criticism of his or her extravagance.
Burn the midnight oil - sit up or work late, especially to study
Midnight oil appears to have been coined by Francis Quarles (1592-1644) in his successful and popular Emblems (1635): 'We spend our midday sweat, our midnight oil, / We tire the night in thought, the day in toil'.
...Other verbs besides 'spend' were subsequently used in adaptations of the quotation, but 'burn' has been invariable since the latter part of the 19th century.
...The expression may owe something to an earlier one of the 16th century: something was said to 'smell of the oil' if it bore the marks of laborious study, i.e. of working long into the night by the light of an oil-lamp.
Bury the hatchet - end a quarrel
An American Indian custom was to bury a tomahawk or other weapon on the conclusion of a peace. The expression is found in writing as early as the 18th century and came into general use by being popularised is such works as Longfellow's Song of Hiawatha (1858).