The themes of "Barn Burning" are related to cultural and socio-economic class distinctions, namely family clannishness. The Southern agrarian tradition holds that family justice supersedes national legal justice. This is why the South seceded from the Union and why Snopes expects his son not to sell him out to the judge.
Another main theme is Faulkner's disregard for "past." He says:
“The past is never dead. It’s not even past.”
“[T]o me,” Faulkner remarked, “no man is himself, he is the sum of his past. There is no such thing really as was because the past is. It is a part of every man, every woman, and every moment. All of his and her ancestry, background, is all a part of himself and herself at any moment.”
Sarty's decision to run away from his family is noble, but--according to Faulkner--the boy can never escape his father's and the South's legacies. They will forever haunt him.
Irony is mainly situational: Snopes' plans to soil De Spain's rug and burn his barn are spoiled by Sarty. Snopes expects family clannishness to win out over social justice, but this, of course, backfires.
Mood is mainly Southern Gothic, with its focus on the grotesque Snopes, fire imagery, soiled rug, and violent ending.