Well this is what happened to him after he died:
…about the body of Saruman a grey mist gathered, and rising slowly to a great height like smoke from a fire, as a pale shrouded figure it loomed over the Hill. For a moment it wavered, looking to the West; but out of the West came a cold wind, and it bent away, and with a cold sigh dissolved into nothing […] long years of death were suddenly revealed in [the body], and it shrank, and the shriveled face became rags of skin upon a hideous skull.
[The Return of the King]
Sauron's death was similar to Saruman's and Gandalf states that Sauron was just going to wander around in Middle Earth, basically forever, as a power-less spirit, so you could assume the same happened to Saruman.
I got that quote from another question that basically answers yours, so I think that counts as duplicate but I'm not 100% sure.