The 100: Jasper and Monty filled this role in Season 1. When they got involved in a more intense and dramatic storyline for Season 2, Wick stepped up as the main funny guy (with a knack for making other characters more snarky whenever they're around him). Murphy also starts to fill this role late in Season 2, serving as the Deadpan Snarker towards Jaha's messianic zeal.
24: Chloe O'Brian is this, as far as a serious show like this allows anyone to be.
Arrow: Felicity Smoak went from one-off episode guest appearance to essentially the female lead thanks to her awkward quips and the actress's comic timing. (Though she also acts as The Smart Guy). As the show started off rather sober in tone without a resident funny guy her handful of scenes significantly lightened the mood (she actually managing to get a smile out of Oliver) and she was brought on as a regular. In an unusual case she's become Oliver's main Love Interest while also remaining the Plucky Comic Relief.
Xander and later Andrew. Though the writing is uniformly witty enough that all characters get their day in the Plucky Comic Relief sun.
This is true with allJoss Whedon shows, but there is still one or two characters in each series who do virtually nothing but be funny, in addition to fulfilling their role in the cast. In chronological order: Lorne on Angel, Wash on Firefly, and Topher on Dollhouse.
Kaylee Did Zoe seem a little, er, funny to you at breakfast?
Wash C'mon Kaylee. We all know I'm the funny one.
Deconstructed Trope with Lorne: the toll that being the Comic Relief takes on him by the fifth season is seriously rough.
Harmony, however, remains her airheaded, self-absorbed self throughout.
Breaking Bad gives us Saul Goodman, who was created for this purpose after Vince Gilligan felt the show was getting too dark.
CSI: Greg Sanders, the DNA specialist, was originally an excellent example of this trope. Later in the series he has several serious character arcs. He also becomes one of the most experienced CSIs on the team. His role has been taken over by Hodges and the supporting Lab Rats.
Doctor Who: Earned a Lampshade Hanging in the episode The Unicorn And The Wasp, when the Doctor calls Donna "The plucky young woman who helps me out":
The Flying Nun: Sister Sisto. In the pilot it was explained that she is studying English sayings in preparation for her eventually teaching in the United States. From that point on her running joke was to say something like "We need to get on the circle!" to immediately be corrected by one of the others "Uh, that's 'ball', Sister."
Lost: Hurley's role began this way, but has become less so as the character has become more important.
Mash: Klinger so seldom got into the dramatic plots of the episodes that his Sarcasm Failure was more effective than Hawkeye's ("Mail Call Three", "Period of Adjustment", "Death Takes a Holiday").
In latter seasons, occasional character Sergeant Rizzo took on this role—his main job being to goof off, sleep, and tell quirky stories about his home in Louisiana.
Monk: Randy Disher. His quirky antics and wild theories really give a nice dose of humor in the show. Still he is semi-competent and is always in the right place when he's needed.
Mr Lucky: Andamo plays this role. Even when being held hostage by a murderer with an itchy trigger finger, he keeps trying to make light of the situation.
The O.C.: Taylor Townsend. As the series Wacky Girl she was a welcome relief from the sometimes overpowering angst of the penultimate season. Usually her wackiness was paired with Seth and Summer's in the aforementioned season to provide comic drama, if such a thing exists.
Shawn Spencer has the distinction of being the main character AND the Plucky Comic Relief. Deconstructed Trope in "An Evening with Mr. Yang," when it's revealed that Shawn needs the Plucky Comic Relief in order to distance himself from the things he has to deal with. When Shawn's mother is kidnapped by the serial killer and Shawn suffers a Heroic BSOD, Gus steps forward to take over the Plucky Comic Relief, much to the confusion of everyone around them.
Woody The Coroner, a Cloudcuckoolander who is there to say and do funny, and sometimes wildly inappropriate, things. In the seventh season premiere ("Santabarbaratown 2"), Woody's the one who provides most of the comic relief because Shawn 's father has just been shot and the situation is too serious for him to make jokes.
Punky Brewster: Allen Anderson was the show's comic relief, although his failure to learn CPR in "Cherie Lifesaver" left him sobered because it nearly took Cherie's life (she was trapped in a disused refrigerator and was found unconscious. Punky and Margaux used CPR learned at school to save her.)
Revolution: Aaron Pittman. He says some pretty funny things, like in "The Children's Crusade", when he compares the group of kids they find to Children of the Corn, and annoying little Ewoks from Star Wars.
Joxer: Now listen, I'm fierce, and I have a lust for blood. As a matter of fact, if a couple of days go by and I haven't shed some blood, I get very depressed. Blood and me go together like a horse and chariot! I once bathed in a tub of blood! My nickname is Bloody Joxer!
Xena: If you like blood so much? Keep talking.
The X-Files: The Lone Gunmen. This did not save them from a random and contrived death in the last season, in an otherwise comedic episode.