Two examples:
Prince’s “Ballad of Dorothy Parker” quotes “…and it was Joni singing/Help me I think I’m falling” - “Help Me”- Joni Mithcell, natch.
Funkadelic’s “One Nation Under a Groove” quotes “so high you can’t get over it/so low you can’t get under it” from The Temptations “Psychedelic Shack”
Not counting rap, because, well just because. 
I’ve thought about starting this thread myself in the past…usually right at the moment when I’m listening to the radio and notice it happening in the song.
The only example I can come up with at the moment is in Stevie Nicks’ Stand Back, when she sings, Like a willow / I can bend" which I take as a reference to the similar line in the Carpenters’ “It’s Going to Take Some Time This Time”, not to mention their reference specifically to willows in “Hurting Each Other”.
Jackie Wilson Said, by Van Morrison.
I don’t know about Psychedelic Shack, but these lyrics are from the old “Negro Spriitual” Rock-a My Soul in the Bosom of Abraham which I learned in elementary school back in the 1950’s
Springsteen singing “Roy Orbison singing for the lonely” in Thunder Road.
Happy Rhodes does it, in her Acoustic Tribute version of the song “Feed The Fire” (the original album version is on her album Warpaint, while the Tribute version is on her compilation RhodeSongs). The song is a thank you to those musicians who inspired her, so in the Tribute version she sings this in the last part of the song, quoting Yes, Kate Bush, and David Bowie:
She also quotes Sheila Chandra and Monsoon’s “Ever So Lonely” (those words) in her song “The Chosen One.” The words themselves aren’t special and some might say it could be a coincidence, but she was a huge Sheila Chandra fan and she told me those words were a tip of the hat.
Warren Zevon does it a little in “Play it all night long”
“Sweet Home Alabama
Play that dead band’s song
Turn the speakers up full blast
And play it all night long.”
Rush’s song “The Spirit of Radio” quotes from Simon & Garfunkle’s “Sounds of Silence:”
“For the words of the prophets were written on the subway walls”
*Well I heard mister Young sing about her
Well, I heard ole Neil put her down
Well, I hope Neil Young will remember
A Southern man don’t need him around anyhow *
Sweet Home Alabama, by Lynyrd Skynyrd in response to Southern Man by Neil Young.
Bowie quotes “A Day in the Life” in “Young American.”
There’s some song I occasionally hear on the radio in the laundromat–which is about the only place I hear any current pop music–that quotes five lines from Dylan’s “Idiot Wind” (“They say I shot a man named Gray/And took his wife to Italy…”) as well as mentioning “Tangled Up in Blue.”
Glass Onion by the Beatles quotes other Beatles’s song titles.
Most, if not all of one entire verse of Thank You (Falettinme be mice elf agin) by Sly and the Family Stone consists of lyrics of Sly and the Family Stone hits:
(I’ve bolded those I know of, and it’s possible there’s more.)
The Beatles in “Come Together” sort of quoted Chuck Berry’s “You Can’t Catch Me”:
Here come old flat-top he come
Groovin’ up with me
(In Berry’s song, it’s “Here come old flat-top he come cruisin’ up with me”)
And speaking of Chuck Berry, his “Brown-Eyed Handsome Man” gets quoted in John Fogerty’s “Centerfield”:
A-roundin’ third, and headed for home,
it’s a brown-eyed handsome man
Close. Rush changed “subway” to “studio” – since their song is about, well, radio.
Hootie & The Blowfish- “I Only Wanna be with You”
Spoke- minor nit- the Beatles line is “here come ol’ flat top/he come groovin up slowly”
The Commodores Night Shift quotes Marvin Gaye “Aw talk to me so you can see
What’s going on” and Jackie Wison “Your love it lifted us Higher and higher”. Not surprising as the song is a tribute to the two fo them.
The Beatles’ “All you Need is Love” quotes “She Loves You, Yeah, Yeah, Yeah”.
One of the songs on “Dream of the Blue Turtles” (sorry, I can’t remember which right now and Google is failing me) quotes the Police song “Every Little Thing She Does is Magic.”
Several rock songs quote Chuck Berry’s line “Meanwhile, I’m still thinkin’” from “Little Queenie.” “Get It On” by T. Rex is one, but I know I’ve heard it elsewhere, too.
Yeah, that’s the one. Thanks.
More Dylan quotes: in Simon & Garfunkel’s “A Simple Desultory Phillipic,” Simon says “But it’s alright, Ma–everybody must get stoned!”
Jaime Brockett’s “Legend of the U.S.S. Titanic” quotes “Like all tough sailors do when they’re far away at sea” from “Bob Dylan’s 115th Dream.”
Quoting other songs is something of a running motif in Genesis’s The Lamb Lies Down on Broadway album, which refers to the Drifters (“They say the lights are always bright on Broadway”), Bacharach-David (“Raindrops keep falling on my head”), the Beatles (“It won’t be long, yeah”), the Stones (“It’s only knock and knowall but I like it”), and possibly others I’m not remembering offhand. (And that’s not even getting into poetic quotations.) The “On Broadway” line is also quoted in David Bowie’s “Aladdin Sane.”
X - The New World “Don’t forget the motor city”
The New York Dolls - Trash “Ah how do you call your loverboy?”