The phrase is, of course, from one of the good Pink Floyd songs: Another Brick In The Wall. I always wonder what could it possibly refer to in the context of the song and in general context. I mean, what kind of exchanges would you term 'Dark sarcasm in the classroom'?
(By the way, there are actually three songs called "Another Brick in the Wall"; the one you're referring to - which starts with "We don't need no education" - is Part II. ABitW, Part I starts "Daddy's gone across the water, leaving just a memory" and Part III starts "I don't need no arms around me; I don't need no drugs to calm me.")
ABitW,Part II is set in a Dickensian/Orwellian nightmare vision of an English boarding school. The teacher has just caught young Pink writing poetry, and he mocks Pink for his literary aspirations by waving the "little black book with poems in" over his head, crying "A poet! The laddie fancies himself a poet!" and then reading the poem out loud, in the most dismissive way possible. (The poem he reads is "Money", from The Dark Side of the Moon.)
In short, "dark sarcasms" refers to the ways in which (bad!) teachers find, and hold up to public ridicule, the weaknesses of their students in order to crush their souls and dreams.
Sarcasm is frequently encountered in everyday life, and it doesn't especially have to be negative - sarcasm is used a lot in humour. Correspondingly, sarcasm is found every day in schools, used by pupils and teachers alike.
However, in the very Orwellian 'Brick in the wall', the 'dark' is meant to convey a more cynical, sinister or particularly ill-intended sarcasm demonstrated by the the teachers; a particular pernicious sarcasm that poisons the minds of the pupils, or (as I think the song intends) the conditioning of the populace by the establishment/government machine.
"Sarcasm" is defined as "—
mocking, contemptuous, or ironic language intended to convey scorn or insult"
"Dark" used as an adjective has several meanings i.e. "dark period in our history" means that period of history was rife with ignorance.
"Dark" however, in this case means " sinister; evil" as in a dark purpose
So "dark sarcasm" can be defined as below:
A taunting, sneering, cutting, or caustic remark; gibe or jeer, generally ironic phrase.