The most common and most likely answer to this crossword clue is the 4 plus 2 letter word STAY UP
I've seen this crossword clue in The Daily Telegraph
S
T
A
Y
U
P
More clues leading to the result STAY UP
- Remain standing to drink, taking in Flower of Scotland
- Remain at Oxford maybe and don’t retire
- Don’t go to bed — party wildly with us right away!
- Refuse to sleep with guy at university
- Don’t go to bed with hero to crash into house party
- Don’t go down river during drink
- Don’t retire, as those who keep from relegation do
- Burn the midnight oil maybe in period of residence at uni?
- Don’t retire on lousy payouts when nothing’s left
- Be in the loft and remain awake for long
Buddy explains!

The straight part
We know that "Burn the midnight oil maybe" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case burn the midnight oil maybe = stay up.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.