In Faulkner's "Barn Burning," the plot is straightforward, but the motives of the characters (especially Abner and Sarty) are complex. In your opinion, why does Abner burn barns and destroy the rug?

Since the story is told primarily from Sarty's point of view, readers don't get much of a glimpse into Abner's thoughts. However, Faulkner does give some clues as to why Abner Snopes burns barns and why he soils the de Spains's rug.
When the Snopes family stops to camp for the night on the way to the de Spain property, Sarty thinks about the tiny fires his father makes whenever they camp. This causes him to contemplate his father's relationship with fire. Faulkner writes that if Sarty had been older, he might have realized that Snopes viewed fire as "the one weapon for the preservation of integrity." This implies that when Snopes feels he has been belittled or disrespected, he retaliates by burning barns as a way to preserve his sense of worth.
Snopes's comments upon leaving the plantation house shed light on why he soiled the rug. After commenting on how "pretty and white" the house is, Snopes says,

"That's sweat. Nigger sweat. Maybe it ain't white enough yet to suit him. Maybe he wants to mix some white sweat with it."

This suggests that Snopes believes the de Spains's wealth has been achieved by oppressing others. He believes the de Spains have used other men's labor to create a comparatively extravagant lifestyle. To punish them for their wealth and their lifestyle that puts them above manual labor, Snopes damages their property.
Snopes acts out when he feels disrespected; he believes burning barns and damaging the property of others helps make up for his poor lot in life and helps to even the score with those who are more successful than he.