When you do stay up way too late and do Duolingo lessons. This can refer to you doing too many lessons and staying up late as a consequence or doing lessons only moments before midnight.
Person A: Why is Jenny on her phone at the party?
Person B: It's almost midnight abd she doesn't wanna lose her streak. She's burning the midnight owl.
Person C: Why are you still awake at 3 am?
Person D: I got hung up on my Doulingo lessons. You know, burning that midnight owl.