The most common and most likely answer to this crossword clue is the 9 letter word CAUTERISE
I've seen this crossword clue in The Daily Telegraph
C
A
U
T
E
R
I
S
E
More clues leading to the result CAUTERISE
Buddy explains!

The straight part
We know that "Burn" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case burn = cauterise.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.
What does CAUTERISE mean?
CAUTERISE verb- To burn, sear, or freeze tissue using a hot iron, electric current or a caustic agent.