Wasn’t that Orson Welles? Because before you posted this, I was looking up the wiki page for the album because I couldn’t remember who did the spoken parts, only to learn to my surprise that there are two versions: the 1976 original which doesn’t contain any spoken parts, and a remix/remaster from 1987 that contains musical overdubs and adds spoken parts by Orson Welles. There is nothing on the wiki page about contributions from Vincent Price, though.
Leonard Nimoy’s “Pure energy” in What’s On Your Mind? by Information Society.
I won a “War of the Worlds” promo jacket when that album came out because I was working at a record store at the time and we won a store - display contest by recreating the cover art in styrofoam and random stuff we had around the back room.
I swore it was Price (and, it sounds like him), but it looks like it was Welles.
William Shatner speaks and yells the entire song of Common People while Joe Jackson sings the chorus.
Jim Carrey provides the voiceover on The Weeknd’s “Phantom Regret by Jim.”
Not sure whether this counts, but on Brent Spiner’s performance of “It’s a Sin To Tell a Lie”, not only is he backed up vocally by Michael Dorn, LeVar Burton, Jonathan Frakes and Patrick Stewart (“The Sunspots”) but Stewart also speaks a verse in the middle of the song.
Primus’s Tommy the Cat, featuring Tom Waits as the titular cat:
From Quadrophenia: BBC’s Jon Curle’s news blurb at the end of “Cut My Hair”.
(It’s a shame there’s no video that lets this segue into the great opening riff of “The Punk And The Godfather”)
The song Pope by Prince features a voiceover from Bernie Mac at the beginning that says, “Y’all don’t understand…I ain’t scared of you motherfuckers” followed by Mac saying “stir it like motherfucking coffee.” It also contains Bernie Mac saying “seek and destroy” partway through the song.
She wasn’t famous, but for her voice-over whispering “Big boys don’t cry” on 10cc’s I’m Not In Love, Kathy Redfern became a minor celebrity. She was actually the secretary at the recording studio.
It was Orson Wells, quoting Poe’s Marginalia XVI:
Gordon Sinclair, Canadian journalist, writer, and commentator.
He was very well known in Canada.
I still get emotional listening to this recording. Especially because it’s from a Canadian’s viewpoint.
Bandleader Archie Bleyer provided the spoken “Yes?” on the Chordettes’ recording of “Mr. Sandman.”
Side 2 of the Small Faces’ Ogdens’ Nut Gone Flake LP from 1968 features the story of Happiness Stan interlinked with narration between the six songs provided by comic monologuist and performer Stanley Unwin in his unique, nonsensical private language of “Unwinese”.
Adam Driver in “Please Mr. Kennedy.” 
In the Bevis Frond’s “Right On (Hippie Dream)”, poet David Tibet reads long excerpts (editorials and letters to the editor, I think) from the famous 1968 psychedelic zine Gandalf’s Garden. The song probably isn’t all that famous among the general public, though it was frequently remarked on by the album’s reviewers.
When I first heard the song, I thought the spoken-word pieces were supposed to be some sort of parody. I only later learned that the texts were apparently written in earnest back in the 1960s.
Cheech Marin and Tommy Chong on Joni Mitchell’s version of Twisted
Wait, no driver on the top?
Man, the chick is twisted, crazy!
Speaking of Pink Floyd, there is the following dialogue from The Wall:
Marshall Dillon: Well, we got only about an hour of daylight left. We better get started.
Miss Tyson: Is it unsafe to travel at night?
Marshall Dillon: It’ll be a lot less safe to stay here. Your father’s gonna pick up our trail before long.
Miss Tyson: Can Lorca ride?
Marshall Dillon: He’ll have to ride. Lorca, time to go! Chengra, thank you for everything. Let’s go.
Miss Tyson: Goodbye, Chengra!
Chengra: Goodbye, Missy!
Miss Tyson: I’ll be back — one day.
Chengra: The bones have told Chengra.
Miss Tyson: Take care of yourself.
Chengra: Marshall, look after my Missy.
The identity of the actors portraying these characters is left as an exercise for the reader.
Love it! Very Bowie-esque. And not a hint of ska.