One of the many running gags in Kim Possible was the "puppy dog pout◊". While it doesn't look the same, it's used to the same effect, and it apparently works on every non-villain she knows; besides, Kim has too much self respect to use it on a bad guy. Her mother, Ron, and Rufus have all appropriated it at one time or another.
Not to forget the subversion aspects: in more Troped shows, puppy dog eyes is one of the tactics a cheerleader would be expected to use to get anything she wants, especially from the parents. In Kim Possible, not only does this work seemingly on anyone, it more specifically works when a mother uses it on her daughter.
These aren't actually exceptions to the above rule — they were on the same side at the time.
In Wolverine and the X-Men Toad of the Brotherhood breaks out the most pathetic cutie-face an anti-hero has probably ever worn when Quicksilver refuses to rescue him from mutant-jail. The large eyed pout is accompanied by a small-voiced "You... you're dumping me? But... but why?"
In "Plane Pals", the Warner siblings use puppy dog expressions on Ivan Blowsky, causing him to feel bad about yelling at them...for about half a second, whereupon the Warners switch to making grotesque faces at Blowsky.
In Teen Titans, this is taken a step further. Beast Boy can morph his entire body into that of an adorable kitten so he can do puppy dog eyes with his entire face◊
Johnny Test parodies this when the title character gives multiple people this look, which is referred to as simply "The Look", with a sad song on a tape player.
Yin and Yang use this to get master Yo buy them stuff in Yin Yang Yo!.
T.U.F.F. Puppy: Dudley does this often. Since he's an actual puppy, it's funnier that way.
In an episode of Phineas and Ferb, Dr. Doofenshmirtz's evil plan of the day involves giving himself big, sad eyes in order to emotionally manipulate the citizens of Danville into giving him what he wants. They work, but only in the dark where his eyes look huge and sad. When the power comes back on they look swollen and bloodshot and have zero effect on his brother (the mayor), other than creeping him out.
The Fireside Girls from use their "big pleading eyes◊" as a way of getting people to buy their cupcakes.
Subverted in Ren and Stimpy - Ren's eyes are already bugged out of their sockets; this takes it a few disturbing steps further.
Stimpy can do it better; just watch the pilot for proof!
"Don't do that too long, or your face might stick that way!" The faces of Precious Moments figurines and the characters of the five Precious Moments animated specials did stick that way. Not only are they stuck in a Super-Deformed proportion, but the upper eyelid is at such an angle that the shape of the whole eye suggests a teardrop, and the iris and pupil are large enough that the white shows up only at the bottom of the teardrop.
Done by Bugs Bunny (towards the audience) in What's Opera, Doc?, after he hears Fudd singing "Kill the wabbit".
June will do this in KaBlam! sometimes to get what she wants.
In an episode of King of the Hill, Hank's friends are trying to find out why Alamo Beer isn't in stock and Peggy knows the secret, and Hank says "Aw Boomhauer don't give me those basset hound eyes" and Boomhauer has his usual expression.
Apple Bloom: "Aren't you gonna stay for brunch...?" See this vid.
Pinkie Pie does this too: "Oh no! My friends don’t like my parties, and they don’t want to be my friends anymore!" See this short vid.
Sweetie Belle deliberately invokes this. "If you don't want to spend time with me..." See this vid.
As of season 4, Fluttershy has started showing a tendency to use these to get her way from time to time. For example, during "Filli Vanilli", she uses them to convince Big Mac to let her be his voice for one last concert. and later, on "Trade Ya", she and Rainbow Dash (of all ponies) try this to convince a trader to go back to his stall to speed along a trade in a long, strange Chain of Deals. In this case it fails, however, as the trader is more concerned about filling his belly.
Spike of Tom and Jerry fame gives his son Tyke a few lessons about being a dog in one short. The first lesson is being "man's best friend", which consists of the begging position and lying at the master's feet. Both "with the big, sad, soulful eyes."
As he does in the movies, this is Puss' specialty in The Adventures of Puss in Boots. His mastery of the technique has given him an extremely high resistance to it, and anyone who tries it on him just causes him to do it back to them to demonstrate how it's really done.
Melissa on Milo Murphy's Law has "doe eyes," which are very effective, but which she says are for "emergency use only."