Context #1: Ted and Wilma just finished an exam…

Ted: How was the exam?
Wilma: I think I did well, but I’m so tired, I don’t know for sure.
Ted: Really? What happened?
Wilma: I was getting ready for bed when I discovered that I didn’t read three chapters of the text book.
Ted: Oh no!
Wilma: Yes, chapters 16, 17, and 18! So I had to burn the midnight oil to catch up. I stayed up until 5am studying those chapters.
Ted: Uh, Wilma, I guess you were too tired to realize…those last chapters weren’t on the exam.

Context #2: Rhonda and William live in a dormitory.

Rhonda: I’m going to bed now. Goodnight, William.
William: Goodnight, Rhonda. Hey, where is your roommate?
Rhonda: Wendy? Oh, she’s at the library burning the midnight oil.
William: Wow! You mean she hasn’t finished her essay yet? That’s unfortunate.
Rhonda: Not for me! I’ll sleep well; I won’t have to hear her snore all night!

Meaning: To work all night. We use this idiom to emphasize the idea that a person is working hard after his or her normal work hours. In the old days (before electricity), people burned oil in lamps in order to light a room at night. If someone is “burning the midnight oil,” it means they need light in order to finish their work at night.

https://languagesystems.edu/