In episode 8x08 of The Vampire Diaries, Damon kills Elena's replacement, Sybil, and throws away Elena's necklace in high definition. This episode focuses on the aftermath of Sybil's control over Damon and his actions while under her influence. 

Damon ultimately chooses to kill Sybil in order to free himself from her hold. The significance of him throwing away Elena's necklace represents his attempt to let go of his past with Elena and move forward. This decision has major implications for Damon's character development and relationships with others in the show.