Two (or more) characters, for whatever reason, must pretend they do not know each other over a long period of time. There are two obvious reasons for this to happen:

1. A spy mission or con game is in progress. If this is the first time some of these characters have appeared on the show, the audience might not be told that the characters know each other either.

2. One character denies knowing another because they think the association would reflect badly on them. This might be a matter of life and death, for example if the person being blanked is an official Unperson and the one doing the blanking is scared of a visit from State Security. In a Teen Drama or Soap Opera, however, it's more likely that one character is just embarrassed to be seen in public with the other, and demands that they pretend not to know each other.

Examples of We Do Not Know Each Other include:

Type 1:

Film

Lucky Number Slevin: The audience isn't told until the end that Slevin and Mr. Goodkat are actually working together in the same con.
Rounders: Mike and Worm use this trope when they work together at the same poker table several times throughout the movie. This backfires in a major way during their final game, where Worm gets caught cheating when dealing a winning full house to Mike, thereby pissing off a room full of New York state cops.
The Phantom Menace had a brief shot of someone on Tatooine who was nabbed by the Star Wars Expanded Universe and made into an ultra-conflicted Jedi named Quinlan Vos. He could have helped Qui-Gon and company, but he was infiltrating something or other and assumed the same of the other Jedi.
Used in one of the western films in the Trinity series, where the titular character Trinity and his brother Bambino pretend not to know each other so that they can cheat at cards. The scheme falls apart when a professional gambler accuses Trinity of cheating, which Trinity was doing by giving everyone at the table increasingly good cards, but his brother ended up with the best hand.
This is the fatal mistake Fredo makes in The Godfather, Part II. He pretends not to know Johnny Ola when they meet in Cuba, but later on babbles excitedly about the various places in Havana that Ola took him to, while Michael can be seen covering his face in despair.

Literature

In Dan Abnett's Warhammer 40,000 novel Brothers Of The Snake, after he worked with the Damocles squad of Space Marines on a mission on Ceres, the Inquisitor Mabuse meets up with them at a coronations. He draws off their leader to alert him that he is passing as a trader and not as an Inquisitor.
Agatha Christie's N or M? has a retired (former British Intelligence) couple going undercover at a hotel. Their respective cover stories were devised separately, so they could not act as though they knew each other.
In The Sookie Stackhouse Mysteries, Eric often follows Sookie into the territories of other vampires. Since, as an important leader in his own territory, he is often not welcome elsewhere, he frequently assumes a false identity for that time.
Harry Potter and the Order of the Phoenix presents an interesting variation: members of the Order working in the Ministry of Magic, such as Arthur Weasley and auror Kingsley Shacklebolt pretend they don't know each other to avoid arousing suspicion, and act rather coldly to each other at work, though Arthur whispers what his wife's cooking for dinner at the Order's headquarters to him afterwards.

Live-Action TV

In the Star Trek: The Next Generation two-parter "Gambit", Riker is captured by a rogue vessel, which has already been infiltrated by a presumed-to-be-dead Captain Picard. Only here he's known as Galen.
Mission: Impossible, being a con game series, uses this nearly every episode.
Unsurprisingly for a show about a gang of long-con operators, this happens all the time in Hustle.
In the same vein as Mission Impossible, Leverage does this several times.

Video Games

Tommy Vercetti and Lance Vance pull this at a mission in Grand Theft Auto Vice City in order to gain Diaz' trust - and finally kill him off.

Web Comics

Not quite a con, but the The Order of the Stick gang are being cautious, and not advertising that they know each other, when Belkar is in the arena here. Leads to him and V being able to insult each other by way of "introduction."

Western Animation

This was spoofed in The Tick (animation) episode "The Tick vs Pineapple Pokopo," where our heroes keep blowing the secret, but the Evil Minions are too stupid to notice.

Type 2

Anime and Manga

In the manga Sanctuary, the two main characters spend much of the time pretending to be strangers to each other, all the while assisting each other in their plan to shake up all social stratas of 90's Japan society.
In season 2 of Code Geass, Lelouch is stuck having to pretend he does not know Nunnally in order to keep up his charade of not having regained his memories while around Suzaku.

Film

In Lawn Dogs, a woman who had just been making out with Trent later avoids him at a party, due to his outsider status.
Attempted by Johnny in Battlefield Earth, when Terl brings out his Love Interest. Since he's speaking Psychlo, the girl has no idea what he's saying. Johnny goes further and tells Terl that he finds the female repulsive. Terl agrees with the latter, being an alien, but then shows a picture that the girl drew, which is clearly of Johnny.

Literature

In the first Red Dwarf novel, Lister first meets Rimmer when the former is driving a (stolen) taxi cab and the latter hires the taxi to take him to a brothel. For various embarrassing reasons, when Lister is later "introduced" to Rimmer as his subordinate, Rimmer pretends not to know Lister.
Aspects of both showed up in Starfighters of Adumar. Wedge Antilles, pilot and all-around hero acting as ambassador to a world believed to be very recently discovered, finds his old flame Iella from Intelligence there, looking at him without any visible recognition. When Wedge sends one of his pilots to talk to her, she asks how the pilots are doing. Later he confronts her and finds that the New Republic has known about this world for somewhat longer than he'd thought, telling him that she can't blow her cover. Later still he confronts her again, and finds that another reason is pain - they didn't break up so much as drift apart, and she'd never tried to bring them back together, and he'd moved on to other relationships, even though they hadn't lasted. She'd like them to both forget about it. But Wedge loved her too.
A tween-book titled Mom, You're Fired! involves the narrator, embarassed by her mother singing opera in public and "dressing like a Gypsy," lying to a new friend and claiming her mother is her babysitter. When she's has her friend over, she asks to call her mother by her first name, on the excuse that it sounds more grown up, to try and preserve the lie. She eventually realizes that her mother is not as embarassing as another friend's mother, (whom she previously wished was her own mother,) but the narrative ends before we see her new friend's reaction to being lied to.

Live-Action TV

In Huge, Chloe spends half the season pretending Alistair is not her brother, since she resents him being at a camp she previously considered her sanctuary.
In Degrassi Junior High, Stephanie (Alpha Bitch) orders her nerdy little brother to pretend that they're not related. When she gets her comeuppance in the first season finale, she apologizes to him in private, and promises to spend more time with him. But when he asks for a few favors, she grins and says, "Don't push your luck."
In Scrubs, JD and Turk pretend not to know each other for a few weeks due to an embarrassing memory that they unwittingly brought up. It was done as a gag though and they're seen talking to each other again a few scenes later.
In an example that fits both types of the trope, when Dr Ethan Pierce first appeared on Shortland Street he and Dr Brooke Freeman pretended not to know each other, even though they had been lovers for years. Brooke felt that Ethan was an embarrassing reminder of her past and was already dating another man, while Ethan decided to go along with the ruse in order so that he could also date other women as well as to use Brooke as a spy to gain a professional advantage at the hospital that they both work at.
In a recent Season 3 episode of Gossip Girl, Blair bumps into Dan and Vanessa and insists on this trope. Though she then backtracks when she realises how easily Dan is fitting into college life and asks him to be her date for Georgina's party
The dramatic variety occurs in Colditz - as the final days of World War II approach, the prison camp's Nazi Political Officer calls his girlfriend to ask her to run away with him. Scared that the phone might be tapped, she tearfully denies knowing who he is, and says he must have a wrong number.

Religion

In The Bible, Peter denies knowing Jesus when he's brought to trial the night before the crucifixion, making this Older Than Feudalism. And it was just a few hours after Peter said he would never do such a thing. And Christ being all-knowing, insisted he would, and he did. Peter was facing trial and horrible, excruciating death rather than embarrassment, but still...

Video Games

In Dissidia 012, Cloud denies knowing Tifa when she asks because they were summoned on opposing sides.

Western Animation

In Daria, snotty Quinn is terrified that having Daria as a sister will endanger her social status—so the two must pretend not to be sisters. In a late episode, Quinn finally softens and admits this to her friends, who reply, "Oh, we knew. We were just being polite."