Legendary comedy team Bob & Ray based their entire act around this trope, as applied originally to the medium they worked in, and later expanded to take in every media trend and fad going. Given their influence on modern American comedy - including but not limited to Bob Newhart and George Carlin - it could be argued that they played a major role in popularising the concept.
Bob:(introducing a "human interest" segment) We've found that you listeners enjoy hearing these pathetic people tell their tragic stories.
First Officer Douglas Richardson in Cabin Pressure. It's one of the things he does best, and he's good at everything.
Ford: How would you react if I said I'm not from Guilford after all, but from a small planet somewhere in the vicinity of Betelgeuse? Arthur: I don't know. Why, do you think it's the sort of think you're likely to say?
More obviously, Zaphod snarks at Arthur, Ford snarks at Zaphod, Trillian snarks at both of them and Marvin just snarks.
The late Humphrey Lyttleton on I'm Sorry I Haven't A Clue, whose chairman persona believed the entire game was insane, and the teams were clearly idiots for doing these things. (Except Mornington Crescent. That was Serious Business.)
The current host, Jack Dee, has carried on the tradition, with a few subtle differences. Humph was more bored, Jack is more bitter.
Paul Darrow (best known as Avon in Blake's 7) portrays one of these on Jack FM. He supplies all the recorded between song/advert/news/weather etc voiceovers in this style. (Jack FM is a "jukebox" style station with no DJs apart from a weekday breakfast show.)
"Plans for CSI:Basingstoke were dropped after it was discovered no-one has any dental records and they all share the same DNA"
The Devil in Old Harry's Game is very definitely one of these, though most of the other characters get their moments.
Our Miss Brooks: Miss Brooks' character uses sarcasm with frequency (as does many of the characters she plays in other works).
Everyone in The Phil Harris Alice Faye Show has their moments, but Julius the grocery boy takes the prize. This trope defines how he enters a room and greets Phil and Frankie.
In Riders Radio Theater, a great many of Woody Paul's lines, despite an earnest delivery, seem to absolutely drip with irony.
Spock in the Spock Versus Q audio plays Armageddon Tonight and Did I say that?. Technically, Q is the Deadpan Snarker for most of the latter play, but only because he's Spock. It Makes Sense in Context.