Hello My Baby
Come back to Eireann
I’m Just Wild About Harry
Throw Him Down, McClosky
Please Don’t Talk About Me When I’m Gone
The Latin Quarter (with the lyrics “What is Your Order?”)
Shuffle Off to Buffalo
I Only Have Eyes for You
Some of these I later heard elsewhere, but Looney Tunes were the start.
William Tell Overture.
No, not the Lone Ranger part. The “morning” part.
How about all the armed forces Hymns?
“Arkansas Traveler” aka “The Boggy Road To Millidgeville” aka “I’m Bringing Home A Baby Bumble Bee”.
Carl Stalling composed/adapted/arranged the music for the Looney Tunes. Warner Bros had their own staff orchestra who were getting salaries whether they worked or not, so WB let the guys in the Termite Terrace make use of them. If you scroll down from my link, you’ll find a partial list of songs he adapted for cartoons.
One of my favorite parts of my developing opera fandom is being pleasantly surprised every time I’m enjoying a new-to-me opera and having some bit of music come up and thinking, “Oh, I recognize this from Bugs Bunny.” 
Merrie Melodies’ I’m Wearin’ My Green Fedora (for Dora, for Dora); Drink To Me Only With Thine Eyes; I Love to Sing-A, It’s As Easy (As Falling Off A Log).
“The Irish Washerwoman”
“My Gal Is a High-Born Lady” (an old vaudeville tune written by Barney Fagan)
Bing Crosby’s “Rainy Night in Rio”
I can’t help but be reminded of a fairly obscure old Saturday Night Live fake television commercial from the episode that Jeremy Irons did. He was pitching a compilation of classical pieces that had been featured prominently in Looney Tunes. Kind of amusing, if only to get the opportunity to see Irons subverting his typically grave and dignified image by singing, “Kill the wabbit!” along with the accompanying Wagner selection.
“I want to sing-a about the moon-a and the June-a and the spring-a”
Some that I think were based on real songs, but with changed lyrics?
“I’m bringing home a baby bumble bee…”
“La da da da da…abracadbra…”
The boogie-woogie piano from Tom and Jerry (Jerry playing the bass notes, Tom joins in on the high notes)
No Camptown races? I believe that was Foghorn’s favorite.
I’ll just complement that with “Largo al Factotum” from Rossini’s “The Barber of Seville”. The song that Giovanni Jones is trying to rehearse while Bugs plays the quoted song, which Giovanni starts to sing as well.
“It’s Magic”- music by Jule Styne and lyrics by Sammy Cahn
Oh, and they often used the 1920’s standard “If I Could Be WIth You an Hour Tonight.”
(Ricky ticky tucky…)
They always used this one in episodes set in France- I never knew the title, until fairly recently.
Another classic from Harry Warren.
“Those Endearing Young Charms,” except I have no idea what comes after the explosion.
Ah 
the music is one of the reasons I adore old cartoons - I really love the old Betty Boop stuff with Cab Calloway.
“Sleep, Baby, Sleep”