I recently came across this line:
[Father to son upon seeing how messy his room is]
—Так, пылесос в зубы, и чтоб через час здесь был порядок как в казарме!
Then, researching this idiom online further, the only other example I found was this one:
Дать им лопаты в зубы, и пусть весь огород вскопают.
The only interpretation I can imagine is that (figuratively) putting something between your teeth keeps you from protesting, so perhaps this phrase gives the imperative or the situation a nuance of just shut up and do it! (??)