Well, he didn't actually die (since he was a Maiar, like you said), but his spirit separated from his body much like Sauron's after the Downfall of Númenor. As an incorporeal spirit, he should have been called to the Halls of Mandos, but the tale implies that he was barred from returning. Tolkien indicated that his spirit was left naked, powerless and wandering, never to return to Middle-earth:
"Whereas Curunir was cast down, and utterly humbled, and perished at
last by the hand of an oppressed slave; and his spirit went
whither-soever it was doomed to go, and to Middle-earth, whether naked
or embodied, came never back"
I found an interesting discussion here.
Also, from Return of the King:
…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.
That was more about what happened the moment he died. (When Tolkien refers metaphorically to the grey mist appearing like a pale shrouded figure gazing almost imploringly to the West), it is plain that a great wind, like the pounding of a judge's gavel, passed final judgement on both of them and dispersed their spirits forever. Unlike Morgoth, whose spiritual and/or physical manifestation was imprisoned until the final battle at the end of all things, Sauron and Saruman would remain incorporeal and impotent, and would no longer plague Middle-earth.
TL;DR: Yes, he continued to exist as an ethereal Maiar after he was murdered, but he was without any power and he was doomed to wander but never to return to Middle-Earth