The most common and most likely answer to this crossword clue is the 4 plus 2 plus 6 letter word MAKE IT SNAPPY
I've seen this crossword clue in The Times
M
A
K
E
I
T
S
N
A
P
P
Y
Buddy explains!
The straight part
We know that "Be quick/teasing the dog may do this" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case be quick/teasing the dog may do this = make it snappy.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.