There has always been a great deal of debate as to where Darl stands in the narrative of As I Lay Dying; you could make cases for him playing many roles, from protagonist to antagonist to a vehicle for Faulkner's own commentary on the story. There are certainly many arguments that can be made for Darl serving the role of a tragic hero.
Consider, for example, comparing Darl to his brother, Cash. Cash is a man of pure consequence. He is not over-burdened by introspection or emotion, and as we can see from the earliest chapters, he is motivated by utility alone. Darl is completely the opposite. While he is the character with the most emotional depth, he finds this depth to be more of a burden than anything. It certainly serves no purpose in helping him connect with his family.
Darl's clairvoyant nature, his insight, and his ability to articulate not only distances him from his family, but gives him a nihilistic outlook on the nature of his family's mission altogether. This culminates in his attempt to get rid of the corpse by burning down the barn, a decline of character that many scholars have compared to that of Hamlet. Following this dynamic character arc, it is easy to see why Darl Bundren is a tragic hero.
Darl Bundren can be considered a tragic hero largely because his greatest strength leads to his downfall. To make the argument for this point, we will need to look further at what defines tragedy and the tragic hero.
A tragic hero is the main character in a tragedy.
For Darl to be a tragic hero, As I Lay Dying has to be a tragedy.
Speaking broadly, if we look to the definition of a tragic hero, Faulkner's psychically gifted character, Darl, can be argued to fit the general description of a tragic figure, but we will also have to argue that the novel, As I Lay Dying, is a tragedy in order for Darl to truly be a tragic hero.
The Bundren family, and Darl in particular, is confronted by misfortune. The trek to return Addie Bundren to her family's land is rife with mistakes and negative turns. And, finally, Darl is taken away to an insane asylum.
If we define tragedy as "a form of art based on human suffering that offers its audience pleasure" then As I Lay Dying can be seen as a tragedy. (The more difficult argument comes in proving that Darl is the main character in the novel.)
One more necessary characteristic of the tragic hero is that he must not bring the misfortune on himself through vice or weakness, but through a lack of judgment (and often as a result of his strength):
the hero's misfortune is not brought about "by vice and depravity but by some error of judgment."
Darl is betrayed by the family that he has worked to protect and to understand. His family does not want to take care of him the way he takes care of them and thus meets his fate.