A couple of Robert Plant’s solo songs from the 90s fade out with lines from Zeppelin songs. “Tall Cool One” has a couple “hey hey mama!”'s and a guitar riff from The Ocean. I think he had another solo song that did something similar.
Billy Bragg’s “A New England” borrows Paul Simon’s lyrics, “I was twenty-one years when I wrote this song/I’m twenty-two now, but I won’t be for long” from “Leaves That Are Green.”
Noel Gallagher is the Crown Prince of Quotation - Oasis’s live versions of “Whatever” frequently ended with the Brothers G. singing the “all the young dudes” refrain written by David Bowie for Mott the Hoople. There’s also the melody for “Shakermaker” that was lifted almost entirely from the “I’d Like to Teach the World to Sing” Coke ads from the '70s, “Cigarettes and Alcohol” lifts the riff from T. Rex’s “Bang A Gong” almost note for note… they even went as far as to quote themselves (cf. “come on, come on, yeah yeah yeah” from “Columbia” reappearing a scant three years later in “Be Here Now”).
blink-182 quotes a Screeching Weasel lyric in “Wasting Time” (“in my town you can’t drive naked”) as well.
A particularly egregious (not to mention insulting) example can be found in American Hi-Fi’s “The Art of Losing,” when Stacey quotes the “hey ho let’s go” refrain from “Blitzkrieg Bop” and then the chorus to Kim Wilde’s “Kids In America” is brought in inexplicably in the coda. I’ve never felt so patronized by pop music…
I am assuming the OP excludes blatant ripoffs like “Ice, Ice Baby”?
For a second there, I was thinking, “Gangsta’s Paradise? Weird Al makes songs that sound like other songs? You’re on to something!” 
But I see where you’re going.
I don’t listen to Springsteen much, but I’m told that references to his music crop up a lot, such as in “Cowgirls…”. Titus Andronicus sometimes riff off of Springsteen, but not nearly as much as Gaslight. They do Billy Bragg, too, and I curse you, garygnu. That earworm was almost out of my head, now it’s back with a vengeance.
Bad Religion: “21st Century (Digital Boy)” borrows title and a few lyrics from King Crimson’s “21st Century Schizoid Man.” Their song “Bored & Extremely Dangerous” sounds subtly in part like “A Day in the Life” with the transition, but maybe it’s just me.
Trollish metal bad Stormtroopers of Death have many songs (“The Ballad of …”), where they play a riff and vocal stylings of a deceased musician, ending the song with “You’re dead!”
Almost forgot! Dead Can Dance’s “The Carnival Is Over” (one of the most beautiful songs ever written, by the way) quotes a Joy Division lyric that goes “the procession moves on, the shouting is over.” Not sure which JD song that’s from… anyone?
The Eternal.
Dazed and confused by Led zeppelin and Paranoid by Black Sabbath share a riff.
Indus by Dead Can Dance uses Within You, Without You by the Beatles.
Dang it! Can’t believe I got that wrong.
Hmm. The online lyrics to “My Whole World Lies Waiting Behind Door Number Three” by Jimmy Buffett and Steve Goodman don’t have it, but when I heard Goodman do it in concert he included the lines about “do you want to make a deal” from “Like a Rolling Stone.”
Jamie Brockett’s Titanic song quotes Dylan also “Like all tough sailors do, when they’re far away at sea” from “Bob Dylan’s 115th Dream” from Bringing it All Back Home.
They settled their differences over dinner.
Lennon was sued over that “paraphrase” and agreed to cover You Can’t Catch Me and a few other songs as part of the settlement.
The Beatles’ “Run For Your Life” starts with a line from Elvis Presley’s early hit, “Baby, Let’s Play House.”
I’d rather see you dead, little girl, than to be with another man.
Squeeze in their Loving you tonight (feels good) has an organ riff very similar to Tempted. The song includes the line “my last temptation was misunderstood” so perhaps Squeeze are trying to better explain that song with the new song.
“Hello City” by the Barenaked Ladies directly uses lines from the Housemartins’ “Happy Hour.” The in-joke is BNL was often compared to the Housemartins.
[did we have this thread before…??]
The old 30s blues song “Two Old Maids in a Folding Bed” traditionally quoted from other popular songs of the era (the linked version is one of the cleaner ones*). The lyrics were
Two old maids in a folding bed
One turned over to the other and said . . .
At which point a song quote would be added.
*“Two old maids” was a euphemism for lesbians. Note one of the quotes is from “Yes, we have no bananas.”
The similarity between George Harrison’s “My Sweet Lord” and The Chiffon’s “He’s So Fine” prompted a famous lawsuit.
Another Beatles one: The Raspberries’ “Please Go All the Way” uses the “come on, come on, come on, come on” thing that the Beatles did in “Please Please Me.”
The Veruca Salt song Number One Blind quotes part of Stormy Weather by The Pixies:
It is time
It is time, oh oh
They also reference Glass Onion by The Beatles as well as their own song Seether in the song Volcano Girls with the lines:
I told you about the Seether before
You know the one that’s neither or nor
Well here’s another clue if you please,
The Seether’s Louise
Not a song, but the title of their album American Thighs is a reference to AC/DC’s You Shook Me All Night Long.
Jimmy Eat World’s “A Praise Chorus” nicks lyrics from several songs. I came in to bring up “Crimson and Clover,” but after visiting Wikipedia, I now know that it is one of many.
If you’re going to mention him, you need to also mention his parody of “Safety Dance” by Men Without Hats, which includes the entire lyrics to the Brady Bunch theme song, and his parody of “Money for Nothing” by by Dire Straits, in which the entire song is a modified version of the lyrics of the Beverly Hillbillies theme.
Plus as you may have found on wiki, one of the songs referenced is “Why Did We Ever Meet” by The Promise Ring, and, that part of the Jimmy Eat World song is sung by a guest vocalist, Davey von Bohlen of The Promise Ring.
They tour with Davey’s bands once in awhile, I’d see them just to see him pop up on stage and sing it with them.