if they don’t need to be human… Muskrat Suzie and Muskrat Sam in Muskrat Love.
Wanda and Earl, “Goodbye Earl,” Dixie Chicks
Also Minnie and the King of Sweden.
Judy and Johnny in both “It’s My Party and I’ll Cry If I Want To” and “Judy’s Turn to Cry”.
Spotcheck Billy & Juanita from “Fat Man In The Bathtub” by Little Feat.
“All I want in this life of mine is some good clean fun, all I want in this life and time is some Hit and Run.”
Unfortunately for Rocky, I think the named couple is Dan and Nancy “Lil” Magil.
Middle of the Road - Tweedle dee and Tweedle dum . and the names are McGregor and MacDougal (not tweedle dee and tweedle dum as you might have expected)
Sweet Betsy from Pike (who crossed the wide valley with her lover Ike)
Johnny and Jenny from Pop Goes the World. The whole album is a concept album featuring them.
Ballad of John & Yoko mentions them in the title, but not the lyrics
There’s at least one song referencing Johnny Cash and June Carter, but I can’t think of it now. Probably more than one.
Oh, and Chelsea Hotel by Alejandro Escovedo mentions Sid and Nancy. Apparently Alejandro lived there when that murder went down.
Alice’s Restaurant - Alice & Ray (Brock)
Muskrat Susie, Muskrat Sam …
Which reminds me: “Hello, Shirley? This is Squirrely.”
Johnny & June by Heidi Newfield.
Johnny and Judy in “It’s My Party”
Reprise: “Judy’s Turn To Cry”
Dylan, “I Shall Be Free”:
I make love to Elizabeth Taylor
Catch hell from Richard Burton
Surprised no one has mentioned this, or more likely I missed it. The two young lovers with nothing better to do:
Billy Jo and Bobby Sue - the Steve Miller Band “Take the Money and Run”
Post #30.
Nine times out of ten, when somebody says “I’m surprised no one has mentioned this,” someone has mentioned it. 
Were Pancho & Lefty considered a couple though?:dubious:
Only counts if they are in the lyrics.
Creeque Alley by the Mamas & Papas. It starts with John and Mitchy (Michelle) as the main couple, then adds Denny, then name-checks everyone in the folk-rock music scene, and ends every verse with Mama Cass.