Alexander's Ragtime Band quotes phrases from tunes as well as mentioning them (Swanee River, bugle call).

There's a phrase in the melody of "The Toorie on his Bonnet" where a change of key temporarily evokes bagpipes, but I don't suppose it's supposed to be a particular song.

Songs embedded within long poems also spring to mind for me. Sir Walter Scott put complete song lyrics within long narrative poems (e.g. The Lady of the Lake).

Geoffrey Chaucer's Canterbury Tales contain stories within stories within the story, though the tales are recitations, not songs. Writing in a day when practically all books had to be hand-written, Chaucer crafted his narrative poems with all the mnemonic devices to allow them to be remembered and recited from memory. In this example from the Nun's Priest's Tale, Chantecleer rebukes his wife Pertelote's doubt of his ominous dream (lines 15073-15115; fer=far, hem=them, again=against/next to, wende=go, dreint=drowned, sweven=dream, casuelly=for some reason, reccheles=reckless):


Two men that wold han passed over the see
For certain cause in to a fer contree,
If that the wind ne hadde ben contrarie,
That made hem in a citee for to tarie,
That stood ful mery upon an haven side.
But on a day, agein the even tide,

The wind gan change, and blew right as hem lest
Jolif and glad they wenten to hir rest,

And casten hem ful erly for to saile;
But to that o man fell a gret mervaile.

That on of hem in sleping as he lay,
He mette a wonder dreme, again the day:
Him thought a man stood by his beddes side,
And him commanded, that he shuld abide,
And said him thus; if thou to-morwe wende,
Thou shalt be dreint; my tale is at an ende.

He woke, and told his felaw what he met,
And praied him his viage for to let,
As for that day, he prayd him for to abide.

His felaw that lay by his beddes side,
Gan for to laugh, and scorned him ful faste.
No dreme, quod he, may so my herte agaste,
That I wol leten for to do my thinges.

I sette not a straw by thy dreminges,
For swevens ben but vanitees and japes.
Men dreme al day of oules and of apes,
And eke of many a mase therwithal;
Men dreme of thing that never was, ne shal.
But sith I see that thou wolt here abide,
And thus forslouthen wilfully thy tide,
God wot it reweth me, and have good day.
And thus he took his leve, and went his way.

But or that he had half his cours ysailed,
N'ot I not why, ne what meschance it ailed,
But casuelly the shippes bottom rente,
And ship and man under the water wente
In sight of other shippes ther beside,
That with him sailed at the same tide.

And therfore, faire Pertelote so dere,
By swiche ensamples olde maist thou lere,
That no man shulde be to reccheles
Of dremes, for I say thee douteles,
That many a dreme ful sore is for to drede.