Is casting magical tools and amulets into the Nile after reciting Ayatul-Kursi over them correct or incorrect, and can this harm anyone?

If you believe that what you have is magic (sihr), it is best to bury it, just as the Prophet, peace and blessings be upon him, did when he was afflicted by magic. He commanded that the magic he found in the well of Dhi Arwan be buried. There is no harm in throwing magic into the sea, because like burying it, it leads to its destruction.