I understand "storytelling" is such a broad term in rap, but the songs must include at least some form of running narrative or scenario. I find that when a rapper has a great storytelling song, they're consistently great in this department, so expect to see many from the same artist.
list ordered by
Dance With the Devil Immortal Technique
Stan Eminem, Dido
Brenda's Got a Baby Tupac Shakur
Children's Story Slick Rick
I Gave You Power Nas
'97 Bonnie & Clyde Eminem
You Never Know Jean Grae, Immortal Technique
Jan Rogoszinski added It Was a Good Day Ice Cube
I Got a Story to Tell The Notorious B.I.G.
Niggas Bleed The Notorious B.I.G.
I Used to Love H.E.R. Common, Common Sense
Undying Love Nas
Uncommon Valor: A Vietnam Story R.A. The Rugged Man, Jedi Mind Tricks
Warning The Notorious B.I.G.
I Seen a Man Die Scarface
MaxWilliams added PROM/KING
nocashgoash added Trapped 2Pac
The Heist Big L
nocashgoash added Violent 2pac