Snark-to-Snark Combat

Lucius: ... I've come to explain to you why we're going to have to put our deal on hold. We can't afford to be seen to do business with, well, whatever it is you're accused of being. A businessman of your stature will understand.
Lau: I think, Mr. Fox, that a simple phone call would have sufficed.
Lucius: Well, I do love Chinese food. And Mr. Wayne didn't want you to think we'd been deliberately wasting your time.

Lau: Just accidentally wasting it.

—The Dark Knight

Much like having different hammy actors in the same work, sometimes putting two snarky characters into the same works and having them interact in some form of discussion (especially an argument) can seem like a competition in sarcasm.

Often makes it a little easier to tell which character is more of a snarker.

See also Volleying Insults and World of Snark.

Not to be confused with cruise missile warfare.

Examples of Snark-to-Snark Combat include:

Comic Books

Tends to happen a lot with Nightwing, particularly when going up against Batman himself or Alfred. On that note, many of Batman and Alfred's more casual conversations with each other tend towards this - Alfred usually wins.
Spider-Man and The Human Torch have been doing this sort of thing whenever they team up or are even in the same story together for decades. It's kind of their thing.

Fan Works

Whenever Tora or Sarakshi from Seduction talk.

Tora: Who do you think Hoseki's representatives would believe? One of the maids from a café that has done a grand event for the first time, or the well-mannered and chivalrous Igarashi heir?
Sarakshi: Well-mannered. Chivalrous. Excuse me while I gag, Igarashi.
Tora: I'm sure something as unattractive as that would be right up your street.

Films -- Live-Action

Lucius and Lau from The Dark Knight provide the page-quote dialogue-sample.
The Social Network also has this.
In the Black Comedy Kiss Kiss Bang Bang, Harry and Perry go at it at every turn, with Harry losing most of the battles due to his idiot status as opposed to Perry's cleverness.
James Bond vs. R.
- In The World Is Not Enough...

Bond: If you're Q, does this make him R?
R: Ah yes, the legendary 007 wit... or half of it anyway.

- ... and Die Another Day.

Bond: Give me the old firing range anyday, Quartermaster.
R: Yes, well they call it the Future, so get used to it. (takes Bond into a museum of call backs)
Bond: This where they keep the old relics, is it?

Literature

Much of the interactions between Elizabeth Bennett and Mr. Darcy in Pride and Prejudice.
Half of the conversations in Harry Potter.
Ranger's Apprentice Halt usually wins, so when a younger character (usually Will or Horace) wins, they feel a sense of pride, for some reason.
What happens when The Dresden Files puts Harry Dresden and Gentleman Johnny Marcone in the same room.

Live-Action TV

Gilmore Girls. The scripts of that show do not call for pauses.
Mystery Science Theater 3000. The boys try to out-snark one another on some topic when one of them makes a seminal joke.
Jim Brass and Gil Grissom sometimes used to get into this in CSI.
Niles and C.C. from The Nanny.
Buffy the Vampire Slayer uses this a lot. Lampshaded in Season 4 by a random vamp:

Vamp: Are we gonna fight, or is there just gonna be a monster sarcasm rally?

Castle is pretty much driven by this and Unresolved Sexual Tension.
Most of the main cast of Sherlock gets in on this. Sherlock vs. John, Sherlock vs. Lestrade, John vs. Mycroft, Sherlock vs. Mycroft, etc.
Being a Joss Whedon show, Firefly has a lot of this going on. Virtually all of Mal and Zoe's dialogue, for example.

Zoe: You paid money for this, sir? On purpose?
Mal: Ship like this, be with you till the day you die.
Zoe: Because it's a death trap.

Jeff and Britta from Community are masters of this.
Blackadder and anyone who's as smart as he is. Blackadder usually butts heads with this person, (Melchett, Lord Flashheart, Captain Darling) and wins in the end. Usually hilariously.
Many of the characters of Stargate SG-1 get into this at one point or another, especially between Daniel Jackson and Jack O'Neill in the later seasons:

Jackson: They'll never see it coming.
O'Neill: Which is one of the advantages of a totally insane idea.
Jackson: Yeah, where'd I learn that from?

Chuck and Blair on Gossip Girl, nearly all the time—one of the other characters refers to it as foreplay.
Frasier and Niles Crane on Frasier were the absolute masters of this trope, often doing it very much intentionally and lampshading it in-universe. It stems mainly from their Sibling Rivalry.
All the characters on How I Met Your Mother occasionally indulge in this, especially Ted and Robin.
Dr. Cox and Jordan on Scrubs do this all the time, as part of their Slap Slap Kiss, often to very vicious and/or humiliating levels.
Just about any conversation in Red Dwarf.

Theater

Zeroth Law example: Benedick and Beatrice of Much Ado About Nothing.
The Importance of Being Earnest
- Cecily Cardew and Gwendolyn Fairfax.
- On that note, Jack and Algernon as well.
- And Lane and Algernon... Actually, pretty much every conversation in this play is Snark-to-Snark Combat.

Video Games

Mass Effect 2 cheerfully descends into this in "Lair of the Shadow Broker", when Shepard and Liara snark to each other about game mechanics.

Shepard: Not even a guard rail. I bet the Broker's agents love patrolling the hull.
Liara: At least the view is nice.

Web Comics

Questionable Content likes having people try to out-sass Faye. It never works.
Homestuck: Almost every conversation with Rose and Dave ever. In fact, most conversations between most of the cast lean this way, given how many of them trade in the snarky goodness.
Literal Snark To Snark Combat breaks out toward the end of Errant Story, when Deadpan Snarker Sarine engages in a duel to the death with Snark Knight (or possibly the other way around) Sarna, who also had been Sarine's best friend for centuries. Snarkage mixes with blood and gore right up to the moment someone dies.

Western Animation

Brian the Dog and Stewie Griffin, from Family Guy, are two of the snarkiest characters on the show. As such, conversations between them involve them being especially snarky. Here is an example.

Brian: Oh, there's my laptop. Do you mind? I want to check my email.
Stewie: Go away, I'm editing this music video I am making for Susie.
Brian: Oh really? A music video? Working on a little video there? A little music video? (voice starts to increase in pitch as he goes on) A little compliation of visual images to go with a song? A little 4-minute movie that tells the story of a...
Stewie: Yeah, that only works when I do it.

Pretty much every Battle Couple in Justice League is like this. Green Lantern/Hawkgirl, Green Arrow/Black Canary, Huntress/Question...
This seems to be the only way anyone is capable of communicating on Archer.