Sarcasm Failure is, to put it simply, when a character who you'd expect would be able to deliver an irreverent, sarcastic or deadpan comment on just about anything fails to do so because of the nature of the current situation. This trope can be Played for Drama — if the writers want to show that a situation is particularly dire, having the funny guy go completely serious is one way of showing that — or Played for Laughs, by having the character encounter a situation so completely, utterly, and patently absurd that he or she is rendered temporarily incapable of responding coherently. (Basically their Snark short-circuits.)
Comedy versions may be reduced to gawking, stuttering, or incoherent babbling rage at the sheer idiocy of it all, a Beat, or just greet the situation with stunned incredulity, often lampshaded with some variation of "words fail me".
Not to be confused with Sarcasm-Blind, Cannot Convey Sarcasm, nor with Insult Backfire, though an Insult Backfire may well take the form of a sarcastic insult that fails to be appreciated as such. Compare O.O.C. Is Serious Business, Flat "What", I Have Nothing to Say to That. Compare also Shoo Out the Clowns, when this is an indicator the entire story is taking a turn for the serious. Compare and contrast Dude, Not Funny!, the reaction when someone attempts to be funny about a situation everyone else sees as no laughing matter.