The "slice through ninjas like fruit" bit in Artists vs Turtles certainly counts: the banana in (the artist) Donatello's hand is so clearly Photoshopped that it's translucent.
After Holmes drops his finisher in his rap battle against Batman, Watson can be seen stepping over a railing in the background, revealing that it's actually a greenscreen.
The scene of Bill Nye walking into frame as he says "And I'm still in my prime" doesn't start offscreen, meaning that he literally just blinks into existence rather than walking into frame.
From the same battle, in his title card, when Bill runs across the screen holding a beaker, his shadow is very short and a bit deformed.
Not so much "special effects" as "camera zoom", but it still counts: when Juliet stabs herself, because the camera isn't zoomed in enough, it's very obvious she just thrust the knife beside herself and was nowhere close to her chest. Said knife also completely disappears in the shot where she's on the ground dead.
When Walt Disney draws the Mickey Mouse logo, the sparkles he creates... really don't come very close to following the outline made by his finger.
Any time a small version of a character is shown to be standing on something (Stephen King having two small versions of himself on his hands, Isaac Newton on a book Bill Nye is holding, etc.), it's very likely that they'll look to be standing somewhat off of it, or just otherwise not standing in a realistic way.
Thor's mouth is very obviously not moving when he delivers the line "For thousands of kilometers, discovering the continents". It's especially glaring given that the battle is otherwise pretty good with small details even in wide shots.
During the "Deadpool vs Boba Fett" bonus battle, Boba Fett's jetpack tends to disappear from his body when he's doing certain dance moves. These are most notable when Boba's in the background dancing while in the foreground he's rapping. In his last verse its even worse, as it disappears from the one in the foreground when he spins around then reappears after he finishes.
When Frederick Douglass flips a coin in his battle against Thomas Jefferson, it's absolutely painful to watch it land - the heads side of the coin obviously changes, making it very obvious that the flipping was a separate animation that somehow they didn't consider altering to be consistent with the coin in Frederick's hand.
In a wide shot, Julia Child sets a dogfish on her cutting board. It then cuts to a close-up where she appears to be hacking at it with her cleaver, but when we return to the wide shot, the fish is clearly undamaged.
The mask used to portray Freddy Krueger ends up looking kind of false, the lip-syncing of it is also pretty poor, as sometimes the lips synced well and sometimes they don’t, and sometimes the mouth shows teeth and sometimes it doesn’t.
The overall costume used to portray Thanos ends up looking even worse than Freddy’s. He is too short, most noticeably in the opening shot and his side-to-side scenes with Oppenheimer, where Thanos is clearly simply standing closer to the camera than his opponent. The mask used is both too big in comparison to Lloyd’s shoulders and too small and human-like in comparison to Thanos’ long head, his eyes are not visible whatsoever and his mouth barely moves when he talks. Of course, portraying him accurately might have Snapped their budget...
Peter needed a latex muscle top to play John Rambo - the wristbands hiding the seams is an obvious move, but there's a seam on his neck that's still visible.
Other examples
Parodied in "Superman vs. Goku", as Superman spontaneously grows a gigantic third arm just for the sake of punching Goku as he's dissing the Off-Model animation of Goku's series.
Parodied by Clint Eastwood when rapping against Bruce Lee, when the former's mouth movements deliberately go massively out of sync with what is being said.
The show takes a knock atDoctor Who's tendency towards this. The rap battle is between the Tenth Doctor and Doc Emmett Brown, and Brown makes a comment to the effect of "I don't know what's lamer about you: your fans, or your special effects!" Meanwhile, the TARDIS can be seen flying across the background; not only are the strings visible, but very briefly the hand holding the strings can be seen at the top of the screen.
At the end of the "Godzilla vs King Kong" rap, they deliberately show the motion capture actors for Godzilla and King Kong's dancing at the end of the match.
your fans, or your special effects!" Meanwhile, the TARDIS can be seen flying across the background; not only are the strings visible, but very briefly the hand holding the strings can be seen at the top of the screen.