It is blatantly obvious that "I See the Light" from Tangled was based on "A Whole New World" from Aladdin. Both songs even include the line "It's Crystal Clear" with practically the same notation.
Histeria, an educational cartoon about history, parodied a lot of songs with similar, but original, melodies. In the episode on Teddy Roosevelt, adapted versions of "Wooly Bully" (as "Bully Bully") and "Ghostbusters" (as "Trustbuster") were both used, altered enough to be free of royalties.
Doug's theme song sounds a lot like 'Little Bitty Pretty One' by Thurston Harris.
Doug also used suspiciously similar versions of the Raiders March from Raiders of the Lost Ark, for Race Canyon, Superman's Theme Tune for Quailman, the James Bond theme for Smash Adams, the Mission Impossible theme (in Doug's Secret Song when Porkchop is sneaking into Bebe's house) and a knockoff of "U Can't Touch This" in "Doug Can't Dance" (starts after Mr. Dink stubs his toe), among others.
Jimmy Neutron does this all the time with their montage sequences. One notable one is a couple notes off from "Hey Ya" by Outkast.
Another notable, and rather obscure, one is a knock-off of Raymond Scott's "Powerhouse" (heard in a number of classic Looney Tunes cartoons) in the episode "Return of the Nanobots," during a montage where the Nanobots delete everyone in Retroville (and we even briefly cut away a couple of times to the robots dancing to the music!)
Phineas and Ferb, episode "Ladies and Gentlemen, Meet Max Modem!": Mr. Fletcher, practicing as Max Modem, plays a riff on a synthesizer that sounds a lot like the intro to "Bloody Tears" from the Castlevania series.
The Perry the Platypus theme sounds an awful lot like the song "Secret Agent Man."
"Meatloaf Surprise" features a song that starts off sounding a lot like "Wonderwall" by Oasis, except it's about the singer's love of meatloaf.
In "Meapless in Seattle", the brothers and friends go to Seattle, where they're greeted by a guitar riff that's suspiciously similar to Nirvana's Smells Like Teen Spirit.
The 2012 syndicated version of this show uses a theme sounding suspiciously similar to the Disney Channel version but instrumental.
Chowder has done this twice so far. In the episode "Grubble Gum", where Chowder's gigantic, all-consuming wad of gum starts picking up everything in its path: the background music, appropriately enough, is a lawyer-friendly version of the Katamari Damacy theme. The episode "Sniffle Ball" also has one scene blatantly parodying the classic Super Mario Bros. fight with Bowser, to the tune of a variant on the classic underground theme.
The sleepeating episode uses the first few notes of "Thriller".
Another episode involving Schnitzel's obvious Bruce Lee parody used an equally-obvious and suspiciously similar version of the song "Eye of the Tiger".
And in "Sing Beans", when Schnitzel is playing the steel drum, it sounds EXACTLY like Target's (Australia) "Make Me Happy" aka 100% Happy ad campaign jingle.
There is a recurring theme that often plays when a character is sneaking around, that sounds like the Pink Panther theme. It reuses the first same ten notes.
An early sketch in Robot Chicken features Voltron in a dance-off against a Robeast to "Work It Out On The Floor", an obvious parody of DMX's "Get It On The Floor". On the DVD version and all future airing, the song is replaced with a horribly generic, lawyer-friendly take on the song.
During their "Greatest American Hero" parody, the theme song is "Believe it or don't I'm flying around."
Used in-series when a sketch about the Libertarian Party have them needing to use the suspiciously similar version of various songs because they can't afford the real ones. Such as "Airborn mini-dirt" (Dust in the Wind) and "We are the Victors" (We are the Champions). They still got sued for "Friend choo-choo" (Love Train).
A large amount of the music used for Robot Chicken are suspiciously similar versions of many popular tunes from movies, television shows, and even video games. This makes sense as parodies are a big part of Robot Chicken.
The Teletoon-animated seasons of Johnny Test use their own theme song, which is a Suspiciously Similar Song version of... the original Johnny Test theme song (from when the series was animated by Warner Bros.). The "same franchise as the original music" deal strikes again!
Also occurs on the Cartoon Network broadcast version.
And the Netflix Watch Instantly releases.
Incidentally, while the original Johnny Test theme sounds highly styled after your average top 40 rock band, the second series theme song is actually a suspiciously similar Version of "American Idiot".
One episode of the Angry Beavers featured a suspiciously similar version of "Say You'll Be There" by the Spice Girls.
Futurama does some almost indistinguishable please-don't-sue-us versions of songs like "Happy Birthday To You" and "Santa Claus Is Coming To Town" - the latter has to be listened to very, very closely. The excuse in the director's commentary was 'we thought it'd be alright because the songs would have changed over the years'.
Fry lampshades this during the former by singing "And you smell like one, too!", out of context from the Captain Ersatz version, but in the style of the real song. Of course, that line is not an official part of the song, so there's no copyright violation.
Drawn Together had some songs like this, both on the show, and some on the DVD releases to clear royalties. One DVD example was in the episode "Dirty Pranking Number 2", with the song "This Is Our Version", a song that sounds like "(I've Had) The Time of My Life", the original song that was on the TV broadcast. Another example that occurred on the show was in the episode, "Foxy vs. The Board of Education" with the song "Board of Education", a song that sounds like "I'm Just a Bill" from School House Rock.
Superjail—as explicitly pointed out by the bumps on [adult swim]—could not get the rights to "Love Shack", so the Warden sings the royalty-free, lawyer-friendly, not-gonna-get-sued-for-it "Party Bar" during the episode where they build a bar in the jail.
The episode "Combaticus" features parodies of both "Stone Cold" Steve Austin's theme, as well as the song "Shenhua" from the video game Shenmue.
For The Little Rascals' Christmas Special, a 1979 animated holiday special featuring Our Gang, the producers didn't get the rights to use Leroy Shields' classic theme music, so the special's music composer wrote a suspiciously similar Version of it.
Pretty much the entire soundtrack to the The Powerpuff Girls episode "Meet the Beat-Alls" is suspiciously similar versions of (you guessed it) songs by The Beatles, including "A Hard Day's Night", "Eleanor Rigby", "Across the Universe", "I Want You (She's So Heavy)", "Strawberry Fields Forever", and "The Ballad of John and Yoko".
The finish of "Part of That World" in The Little Mermaid ("Out of the sea/Wish I could be/Part of that world") is Ashman and Menken copying one of their own songs, "Somewhere that's Green" from Little Shop of Horrors ("Far from Skid Row/I dream we'll go/Somewhere that's green").
The episode "Sentries of the Lost Cosmos" from Batman Beyond is full of simulated Star Wars music.
The long-awaited Daria DVD set occasionally opts for suspiciously similar versions in place of the more generic cues that comprise the majority of the music. Technically averted with Cyndi Lauper's "Girls Just Wanna Have Fun" - MTV had apparently tried to license the original recording due to how important it was to the episode it appeared in, but Lauper no longer allows the song to be licensed and they were forced record a cover version (since Mr. DeMartino actually sings the official lyrics).
Sym-Bionic Titan has an interesting version of this trope in the episode "Roar of the White Dragon". When Mike Chan/White Dragon shows up at the school, we hear some music that sounds suspiciously like the song by the Teriyaki Boyz called "Tokyo Drift (Fast and the Furious)". Compare the two here and here. Fitting, but still weird.
There was once a recurring ocarina riff during confrontation scenes, modeled closely on The Good The Bad And The Ugly. And in "One Good Turn", the sting when Bill and Ben meet face to face is basically the Dragnet theme.
While on the subject of trains, the theme from Chuggington shares some similarities with "I Want Candy" by The Strangleloves.
The song "Let Me Be Your Wings" from Don Bluth's Thumbelina is actually a simulated version of the song "A Whole New World" from Aladdin.
The title theme to The Mask is a suspiciously similar song to one of the songs from the movie
Some of the music from Monsters, Inc. are similar songs to "Sing, Sing, Sing."
An episode of Tiny Toon Adventures featured a parody of Super Mario Bros. called Super Pluckio Bros. which played over a suspiciously similar version of the overworld theme of the original Super Mario Bros. game. Adding to this was the fact that sound effects from the actual game were actually used in the episode.
Superman: The Animated Series - its Awesome Music theme bears a striking resemblance to a sped-up version the Aaron Copland masterpiece "Fanfare for the Common Man" melded with bits the John Williams theme from Superman: The Movie. Given that the series was heavily influenced by the John Byrne era Superman (Clark is the real person, Superman is his disguise, and Clark sees himself as a perfectly normal person who happens to have extraordinary powers), this probably wasn't unintentional.
Some folks on YouTube noticed the similarity between The Smurfs theme song and the scatting melody of the J. Geils Band's "Centerfold", both of which came out in 1981.
The episode "Smurfing For Ghosts" uses a tune that sounds very similar to Ray Parker Jr.'s "Ghostbusters", while the cartoon special "The Smurfic Games" uses one that's similar to Vangelis' Chariots of Fire theme tune.
A tune that played during action scenes in She-Ra: Princess of Power contains a 7-note segment that sounds like the theme to Popeye.
The theme to The New Scooby-Doo Mysteries sounds very suspiciously similar to Michael Jackson's "Thriller".
The "This is It!" theme from The Bugs Bunny Show and other television broadcasts of the Looney Tunes shorts sounds similar to the first theme tune from The Flintstones.
There's a background song on Rugrats that sounds a lot like Colonel Bogey March.
The theme to Mighty Mouse: The New Adventures is performed a cappella in a manner silimar to the original Mighty Mouse Playhouse theme, only duplicating the note structure of the line "Here I come to save the day."
In the Ben 10: Omniverse episode "Jolt from the Past", when Ben challenges Rook to find him some opportunities for heroics, the resulting On Patrol Montage is to a tune that sounds a lot like the intro to "Holding Out For A Hero" by Bonnie Tyler.
Largely averted by Regular Show, where the creators typically manage to secure song clearances. In one such notable exception, a soundalike of "Subdivisions" by Rush plays while Mordecai, Rigby and Skips are on a car trip.
The chorus of the theme song to Lilo & Stitch: The Series, "Aloha E Komo Mai", sounds similar to the one of the Mardi Gras song "Iko Iko", except it is a bit sped up and is at a higher pitch.
A rare example of this used with a song in a show itself-the beginning of the song "I'm Skating" from Magic Adventures Of Mumfie sounds like a slowed and pitched down version of "The Beginning of Things"!
The theme to X-Men sounds like Whitney Houston's "I'm Your Baby Tonight".
The theme to Silverhawks sounds like "The Edge of a Dream" by Steven McClintock. Both even have an awesome-sounding guitar solo.
There's some debate as to whether "Let It Go" from Disney's Frozen is most similar to "Defying Gravity" or "No Good Deed", both from Wicked (also starring Idina Menzel, who was the original Elphaba).
The ChalkZone episode "Lost in Chalk" has a sound-alike of "Get Ready for This" by 2 Unlimited playing during the fair...which was recycled from an episode of The Fairly OddParents, "Totally Spaced Out" (Guy Moon, the composer for OddParents also composed the first two seasons and the first half of season three of ChalkZone, and often re-used background music from the former show for the latter)
"The Label Police" includes a sound-alike of Raymond Scott's "Powerhouse".
"Rise and Shine" from Sofia the First sounds similar to "When Will My Life Begin?" from Tangled.
The music playing at Iron Will's seminar in "Putting Your Hoof Down" is actually Eye of the Tiger with one note changed.
In the same episode, when Fluttershy realizes she's become a monster, the Lonely Man theme from The Incredible Hulk starts playing.
During the scene in the episode "It's About Time" where Twilight Sparkle meets a future version of herself, the background music is suspiciously similar to "I am the Doctor" from Series 5 and 6 of Doctor Who.
"One Bad Apple" features an A-Team Montage while the CMC are building the second float, down to the music.
A parody of the Bonanza theme is often used in scenes relating to the Apple family.