Misaki in Futaba-kun Change! drinks steins of honey: not mead (fermented honey), actual beer steins of honey. This plays into her inhuman Sweet Tooth — the girl eats so many sweets that the swim team she's a member of doesn't know how to swim in normal water, since the pool water she swims in turns to sugar water.
Lum and Ten from Urusei Yatsura wreak havoc when they get extremely drunk on umeboshi (pickled plums, usually regarded as a hangover cure). It turns out that alcohol actually cures them of being drunk, though all this is somewhat justfied by Bizarre Alien Biology.
Belldandy from Ah! My Goddess can guzzle down hard liquor like water without even getting tipsy, but one can of soda and she's plastered. What gets a goddess drunk varies for each goddess: Belldandy's sister Urd can only get drunk off of saké.
Carbonated beverages get Mermaids drunk in the manga version of Mermaid Melody Pichi Pichi Pitch. This happens to Lucia while trying to take care of a sick Kaito. In the anime, it's replaced with a magic MacGuffin mood changer that goes haywire.
Panyo Panyo Di Gi Charat has "Inspector Holic" who gets drunk on orange juice!
An odd variation is Charlie Brown from Peanuts, who drowns his sorrows in haircuts.
It's not so much haircuts generically as the fact that his father is a barber. When he's depressed, he goes to his dad's barber shop and, no matter how busy his dad is, he's always happy to see Charlie, and that cheers Charlie up. He's used this to shut up people (usually Violet) who are doing the "My dad is better than your dad at X" thing.
Another from Peanuts: When Snoopy is the World War I flying ace, he sometimes drowns his sorrows in root beer.
Obelix in Astérix sometimes drinks alcohol (Asterix and the Laurel Wreath, for example), but when he and Asterix disagree over the newcomer to the village who's opened a tavern in Asterix and Caesar's Gift, he heads straight to the tavern and orders a goat's milk. Cacofonix comments "If he's drowning his sorrows in goat's milk, he must have had a fight with Asterix". This is because Obelix is strong enough to throw elephants further than Romans can throw javelins, as well as a terrible drunk, and he probably doesn't want to hurt his friends on a drunken stupor, so he's mostly a teetotaler by choice.
A non-Drowning My SorrowsLampshading in one issue of The Avengers: Beast, Wonder Man, Scarlet Witch and Vision are having a night out together, and Beast declares they're going to get "good and sauced". When the waitress points out they've ordered virgin pina coladas, Hank replies "But we've got great imaginations!"
In Iron Man #178, the opening story shows a kid cosplaying as the eponymous hero getting expelled from a group of kids pretending to be the Avengers, and subsequently drowning his sorrows on soda. It's a Breather Episode for the depressing "Tragic Tony Stark" storyline that was going on at the time. You can read it here.
In one of the early, pre-Isodoro Patoruzu strips, Patoruzu goes with Julian de Monte Pio for a night in town, and passes out after only two glasses of champagne. Worse, Julian then reveals that the "champagne" was just apple soda.
In X-Nation 2099, the first issue plays this completely, totally, and literally straight: the characters go to a milk bar, where patches of diodes cause milk to act as a Fantastic Drug.
In My Little Pony: Friendship Is Magic (IDW) Rainbow Dash explains that her memory of the wedding is hazy after having Pinkie's Tutti-Frutti Sherbet Sugar Punch. (It contains seven parts of sugar for every part of fruit!)
The Old West cliche of the gunslinger going into the bar and ordering sasparilla/sarsparilla. (Also, possibly, Getting Crap Past the Radar, since sasparilla was used to treat syphilis)
In Miss Congeniality, we see the heroine ordering a pint of something at the bar. The bartender asks whether she really wants that much. It then turns out to be a pint of ice cream. She even refers to it as getting "chip-faced" (it has chocolate chips).
In Death to Smoochy, Sheldon laments the lack of control over his show by sitting at the bar, a bit drunkenly talking about old kid show hosts... while drinking orange juice.
In Alien Nation, the Newcomers get drunk off spoiled milk. The lactic acid in the fermented milk is produced by anaerobic digestion— just like the production of alcohol.
There's a film about a kung fu fighting toddler who gets drunk on milk.
In A Clockwork Orange, Alex and his violent droogs are first seen with glasses of milk in their hands at their favorite hangout, the Korova Milk Bar. Alex's narration is quick to point out, however, that the milk is laced with various psychotropic drugs to "sharpen you up for a bit of the old ultraviolence."
In Toy Story, when Buzz is having his Heroic BSOD at Sid's house, he somehow manages to get drunk on tea. Imaginary tea.
A Deleted Scene from the 2002 live-action Scooby-Doo film had Velma do a song and dance number after drinking a non-alcoholic drink.
The Further Adventures of Dr A.A.A. McGurk M.D. by Osmar White, opens with a depessed Dr McGurk drowing his sorrows in malted milk at the Explorer's Club. When he comes up with the idea of seeking the Pole of Impossibility (somewhere in the Antarctic) by camel, the other explorers comment "You've put too much malt in your milk."
In Men at Arms, the Watch all Need A Freaking Drink after finding the first victim of the gonne. Three of the six have beer, Nobby has an Umbrella Drink, Detritus has a molten sulphur (because the heat affects troll brains) ... and Carrot has a glass of milk.
Possible example from Harry Potter: Winky the House Elf gets drunk on butterbeer, a beverage usually drunk by underaged wizards without any intoxicating effect, thus having little to no alcohol content. However, house elves are very small individuals (two-three feet tall humanoids) so if the drink really is alcoholic by even a little bit, that would explain how it renders a house elf drunk while doing nothing to teenagers.
Used by Dolphus Raymond in To Kill a Mockingbird who plays the town drunk, swigging from a bottle of Coca-Cola in a paper bag, as a convenient excuse to keep the bigoted townspeople from harassing his family for his marriage to a black woman.
In Gordon Korman's Beware the Fish the main characters slipped some of the school genius' experimental cold cure into the coach's sports drink, unaware that the combination would make him act as if he were drunk. After he wandered over to the girls' school across the road and shocked the headmistress into fainting, they and the genius managed to keep him from further wanderings by playing poker for matchsticks.
Dragons in Jeremy Thatcher Dragon Hatcher are rendered to a somewhat drunken and euphoric state when they are given milk. It makes them very docile and easy to manage. The protagonist does this when his dragon starts getting too large to manage normally.
Live Action TV
In Babylon 5, Vir can be seen drowning his sorrows with a Shirley Temple, a nonalcoholic children's cocktail.
In one Monk novel, Mr. Monk Goes to the Firehouse, Monk gets himself into a hangover of Sierra Springs water.
In an episode of Night Court, when Christine and Roz go to a bar, Christine drinks Roz' signature drink. Typically, Christine makes an ass of herself. Later when she asks "What was in those things?" it turns out it was pineapple juice and soda water.
In the Mystery Science Theater 3000 episode Gamera vs. Zigra Joel and the bots throw a root beer kegger—and get just as drunk as they would drinking real alcohol.
In one episode of Bottom, Richie needs a quick shot of booze to settle his nerves. Eddie gives him a shot of Tizer (a soft drink), which does the trick anyway.
In an episode of Murphy Brown, Miles gets "drunk" off of punch that he assumes must be full of alcohol. There wasn't a drop of anything remotely alcoholic in that punch.
A pub sketch from A Bit of Fry and Laurie has Hugh go off on a long series of inebriated complaints about his wife after having several glasses of ribena (blackcurrant cordial).
Saturday Night Live: The first episode after the 2012 election (hosted by Anne Hathaway) showed Mitt Romney (Jason Sudeikis) actually sneaking milk on the balcony as the news breaks that he has lost the election. Bonus points because Romney is Mormon. And the pop-top sound effect when he opens up the carton.
In an episode of The Golden Girls, Dorothy is sitting in a hotel bar, working up the courage to crash her ex-husband's wedding to his new wife and tell his she still has feelings for him (she doesn't go through with it.) While sitting there, she asks the bartender to "give her another." At first he resists, but when Dorothy insists, he simply says to not blame him when she gets sick, and... puts another bowl of popcorn in front of her.
A placebo effect variation in The Big Bang Theory: Raj is unable to talk to women without consuming alcohol. While on a train in one episode, the boys are surprised when Summer Glau boards, and begin hitting on her. Raj grabs a beer and actually has a great amount of success. Unfortunately, it turns out that the beer is non-alcoholic, and when Howard points this out Raj immediately freezes up again.
Another example of a placebo effect in Grounded For Life; during St. Patrick's Day the Red Butte Pub has ran out of alcohol. Eddie tries to remedy the situation by serving non-alcoholic beer and convinces people they're drunk which works for a while until the customers overhear it's non-alcoholic. One of Brad's friends is convinced he's drunk he starts hitting on Lilly.
Simon from Prozzak drowns his sorrows in milkshakes.
Bye Bye Birdie: After underage Hugo Peabody fails to get served at Maude's, he staggers on the next scene, seemingly drunk.
Mrs. MacAfee: Hugo, what have you been drinking? Hugo: Milk. But it worked!'
When he's trying to get served at Maude's, he uses some terminology that makes it very clear he doesn't know what he's talking about, including asking for a "rocks on the scotch, and put some rocks in it" and a "vodka malted."
Several of the protagonists of Persona 4 get drunk off of non-alcoholic beverages they get at a night club. Unusually, they were just hanging out, with no implications of drowning sorrows- they just happened to get drunk off of regular fruit drinks. It's only Yukiko, Rise and Teddie (the last of whom isn't human and could very well actually get drunk off nonalcoholic drinks) who get hammered, while everyone else is perfectly sober and wondering what happened to them. Even better, the sloshed ones don't know the drinks aren't alcoholic until after they're drunk, and in-story the others speculated that it's either a placebo effect or a result of "the atmosphere". This also happens in the Anime of the Game, except the main character also gets hit by the effects. Or at least pretended to (with him it's hard to tell).
Soda Popinski and his bottles of soda in the Wii version of Punch-Out!!. In the original NES game, it was more of a Frothy Mugs of Water scenario (with his name getting changed from Vodka Drunkenski and just plain old booze), but the Wii remake ran with the soda angle, while hinting that he actually is getting drunk on it anyway.
In the American version of Skies of Arcadia the drink of choice in most taverns is loqua, which is supposed to be simply flavored juice. There are plenty of people around the world that are VERY happy with their juice. In the original Japanese, it's simply wine
Happens with Pepsi in thisVG Cats comic. Leo says it's "like, symbolic and stuff".
Happens on multiple occasions with orange soda in Housepets!
Narbonic's Dave Davenport can get stone drunk on half a glass of vanilla Coke. (And pathos.)
Robin of Shortpacked! (a sugar-powered super speedster) apparently drowns her sorrows by drinking Hi-C and getting on those 25˘ animal-or-vehicle-moves-back-and-forth-for-a-couple-of-minutes rides outside stores.
In the same author's fanfic comics, lactose has the same effect on Kryptonians and Gallifreyans as alcohol does on humans, so Superman and the Doctor really can get drunk on milk.
Homestuck has Terezi and her dangerous reaction to Faygo soda. Her reaction to the overly-sugary soda is akin to an all-out bender, including massive hangovers.
In Captain S, the eponymous captain drowns his sorrows in milkshakes after losing his powers and failing to save videoland, complete with bartender cutting him off.
RandomVerse Spider-Man gets plastered on milkshakes. The first time it's done straight, but ever since then the Hulk and Iron Man have been drugging his milkshakes to get wacky movie-trailer related hallucinations for their own entertainment.
In the same e-mail, The Cheat also gets drunk on soy sauce and tries to eat Bubs' Concession Stand.
Another e-mail has Strong Bad performing a science experiment of the effects of caffeine on Strong Sad. So he slips "a few heaping spoonfuls" of Sanka — instant decaf — into Strong Sad's orange juice, and the few tiny grams of caffeine immediately sends Strong Sad into a caffeinated blitz; you'd think he's inhaled a kilo of cocaine or something.
In one episode of Adventures in Care-a-Lot, Love-a-Lot, bummed out because Cheer apparently forgot her birthday, drowns her sorrows with cups of hot chocolate. After her fourth cup, Funshine asks her if she's had enough.
In The Sponge Bob Square Pants Movie, SpongeBob drowns his sorrows on ice-cream floats at Goofy Goobers. He goes on a bender and wakes up the next morning with a bad hangover (complete with heavily slurred speech and a five-o-clock shadow).
He also passes out from a glass of milk in another episode. (Well actually, this was his "ancestor".)
On Arthur, Buster has drowned his sorrows in chocolate shakes on more than one occasion.
The VeggieTales episode "Silly Singalong 2: The End of Silliness?" opens with Larry, distraught over the cancellation of "Silly Songs with Larry", drowning his sorrows in ice cream. Then he gets a nasty ice cream headache.
In one old Tom and Jerry short, "Blue Cat Blues", Tom goes on a milk binge when depressed.
In an episode of Foster's Home for Imaginary Friends, Mac is depressed and goes to an ice cream shop and orders a sundae. However, being one of those kids who's not allowed to have sugar, he goes completely drunk (a la Spongebob) and has even more ice cream. He leaves with a hangover.
It's more "sugar rush" than "drunk" (he gets manic instead of groggy). It does usually lower his inhibitions and lead him to make remarkably bad decisions, though.
An episode of Freakazoid! had his then-sidekick hospitalized with a minor injury. Freakazoid proceeded to drown his sorrows in papaya smoothies.
Freakazoid: Pour me another one.
Barman: Whoa, slow down there. It's not like Expendable Lad's dead or anything, he's just in the hospital with a bruised clavicle.
Freakazoid: I said give me another one, curse your tiny paper hat! If I wanna blitz myself into some papaya-induced hallucination, that's my business.
Happened in The Simpsons when Bart and Milhouse buy a Squishy that is all syrup, and proceed to get "drunk" from the extreme sugar rush. Bart wakes up the next morning with a hangover, and discovers to his horror that he joined the Junior Campers (Boy Scout expies) during their binge. His sister offers No Sympathy.
Lisa: <tsk, tsk, tsk> The remorse of the sugar junkie.
In the same episode, Milhouse (who also drank that same Squishy with Bart) woke up with a profanity shaved into the back of his head.
Somehow, in another episode detailing the life of Lisa's Saxaphone Mentor, Bleeding Gums Murphy, we find that he spent all of his earnings from his one record on Fabergé Eggs, which is treated like an addiction to alcohol, complete with him telling the man at the store "I'll tell you when I've had enough!".
When Homer and Barney join NASA, Barney excels while sober but becomes instantly drunk with a single glass of non-alcoholic champagne.
Also with Lisa drinking milk at Moe's in "The Burns and the Bees". Moe was reluctant to give her the keys to her bike-lock, he wasn't even sure why he had them in the first place.
They don't get visibly drunk from it, but when the SWAT Kats throw themselves down on the sofa and pop open a couple of cans of condensed milk (being, after all, cats), they certainly treat it as if they are about to down a few brewskis.
On Home Movies, Jason acts like an angry problem drinker when he binges on candy.
In the My Little Pony: Friendship Is Magic episode "The Best Night Ever", Spike is feeling left out after his friends split up at the Grand Galloping Gala, and near the end of the episode he's seen drowning his sorrows... in donuts and hot chocolate.
Spike: Hey, Pony Joe, another donut. Pony Joe: Don't you think you've had enough? Spike: Another donut! EXTRA SPRINKLES!
In "Over a Barrel" we see a pony who's had a few too many salt licks...
In "Super Speedy Cider Squeezy 6000" we see ponies lined up for miles to get a frothy mug ofnonalcoholic cider, showing just how serious they are about their drinking.
Taken to an extreme on Codename: Kids Next Door where sodas and root beer are treated as if they were alcoholic beverages, even sporting an legal drinking age of 13.
Gumby may be the Ur Example, if not the Trope Maker. The little clay boy falls off the milkshake wagon more than once; since he's made of clay, the icy cold confections leave him stiff as a board if he overindulges. "You know what happens if you drink too much ..."
In an episode of Clone High, all of the students get drunk at a party by drinking non-alcoholic beer.
In an episode of Rocko's Modern Life, Heffer finds out that he was adopted, and goes off in search of his real father (who turns out to be a real Jerkass). He finds himself at a diner (clearly a Shout-Out to the famous "Night Hawks" painting), and orders tons of food. (Again complete with telling the man behind the counter, I'll Tell You When I've Had Enough!.)
In another episode, Rocko and Heffer take Filbert to an ice cream parlor for his bachelor party...and wind up with an ice cream hangover (and only minutes to spare to get to the wedding.)
Subverted in the Hey Arnold! movie. When Arnold and Gerald follow the bad guy's goons into a place that says Must Be 21 to Enter, they're drinking a milk in what appears to be a vodka bottle.
In episode 5 of The Legend of Korra, Bolin gets "drunk" on noodles after he sees Korra kiss Mako and gets very sad. The next day he acts hungover, so his "drunkenness" was probably a mixture of lack of sleep, gorging himself, and hysterical sobbing.
Oil is treated this way in Transformers Animated: the Constructicons (as construction worker stereotypes) treat it like brewskis, Megatron sips "fine vintages" of the stuff from a goblet (made from a crushed oil drum), and the Autobots even indulge in festive Christmas "oil nog."
There's an example of this in the episode "In Like Flint" from Ōban Star-Racers. Even though she won the rematch against Grooor, Don Wei said her manner of racing had disgraced the team and they would be dropping out. Molly ran out and the next we see her, she's at the local bar with four empty drink pouches sitting in front of her.
Molly: Another grenadine milk! Pronto! Skrub Bartender: Maybe you should stop. You know what they say, "don't drink milk and drive". Molly: Listen, pal! I already got a father!
In Teen Titans, after the car Cyborg built from scratch is stolen, Raven finds him in a parking lot surrounded by about 15 empty milkshakes, the non-alcoholic version of Drowning My Sorrows.
This is actually possible in real life, thanks to Pavlovian conditioning and the placebo effect. An experiment involved giving free beer at a student party, without telling the students the beer was alcohol-free, of course. Afterwards, the party-goers were questioned. Almost all of them said to be at least tipsy or not able to drive safely any more. They were very surprised when an alcohol test was done and it showed they hadn't drunk any alcohol at all.
In fact, they showed all traditional behavioural symptoms of drunkenness, which cleared up instantly when they found out they were technically sober.
This is also why, in fact, how somebody behaved when drunk can be different depending on the culture; people who believe they're drunk will act like they believe drunks do. Giving people alcohol-free booze just takes it meta.
It's even been observed on This Very Wiki with the My Immortal drinking game anecdotes. If the stories are to be believed, despite the fact that many of the participants are just drinking water, quite a few people got drunk.