For example:
Anthony Di Nozzo, NCIS
Patrick Jane, The Mentalist
Frank Barone, Everybody Loves Raymond
Maj. Frank Burns, MASH*
Carrie Sharples (Martha Raye), Alice
Archie Bunker, All in the Family
Frank Nelson, The Jack Benny Show
C. Montgomery Burns, The Simpsons
Post your annoying characters. 
I generally don’t watch shows with characters that annoy me but I watched Fawlty Towers when it first came out in England. Later when I returned to Australia it was shown here. I loved it.
A couple of years later when I saw it in re-runs it infuriated me. I found Basil Fawlty not at all funny, just the worst kind of insufferable idiot. I fumed at his every action and thought “how does his wife put up with him?” so only watched a couple of episodes.
Years later I saw it yet again and once more Basil was hilarious.
Oh and I should add I love the way Patrick Jane’s dickishness is played for laughs. If it wasn’t for that it would just be another smartass investigator show.
[ul]
[li]Erkel[/li][li]Ray Romano in anything[/li][li]Anyone not named Meg or that sex crazed guy in Family Guy[/li]Dr. Zoidberg[/ul]
Chatsworth Osborne, Jr. was a thorn in the side to both Dobie and Maynard
And who can forget Eddie Haskell?
Kim Kelly from Freaks and Geeks. Hell, Lindsay too most of the time.
Lois Griffin. Her voice keeps getting more and more intolerable.
Crissy Snow. As a kid I thought she was funny, but now she seems shrill and annoying to me.
Does the character have to be from a TV show specifically?
Because I freaking hate that Robot from the US cellular commercial. Hate him BIG time!!!
Jane on Coupling (the original) was completely annoying to me, until the scene with Jake the Truth Snake. Gina Bellman is hysterical; it just made the whole character work from that point on.
Mr. BURNS is who you choose for irritating on the Simpsons? I’d choose Flanders well before Burns.
Ted Baxter, Mary Tyler Moore Show
Q, Star Trek: TNG
Karen, Will & Grace
Roseanne’s mother, Roseanne
Michael Scott on The Office.
Michael Scott on The Office.
Paul Lassiter on Spin City. (And any other role that Richard Kind plays)
Kramer on Seinfeld - heck most of them.
All of them on Everybody Love Raymond.
Everyone on The Office.
Everyone on Will and Grace.
Everyone on 30 Rock except maybe Jack Donaghy.
Everyone on Seinfeld.
Actually come to think of it annoyingness is almost inherent in any interesting character.
Lt Columbo, from (duh!) Columbo, who seemed to make a career of annoying a suspect until they confessed just to get him to go away.
David Spade in Don’t Shoot Me.
Rob Schneider in Men Behaving Badly.
Oz in Auf Wiedersehen, Pet.
Sandra Clark as portrayed by Jackée Harry on 227 is the one answer to this question that I will never forget, even as I forget most of the specifics about that show. I don’t want to kick the actress, she was given a role, developed a character in conjunction with the director … but, damn, that annoying “Maaaawwie”, ugh, why, just why, was that considered entertaining.
Some would add Steve Urkel, I wouldn’t, I think he took annoying and made it charming. But I might be too generous, and I didn’t really watch that show.
Megan Mullally as Karen Walker on *Will and Grace * might also fit the bill. I only saw it for a few moments, but I got a quick feeling that this was another Jackée Harry-esque character. Again, actress was doing what she was supposed to do, I guess, but who decided to write a show around annoying?
Fawlty Towers is a tough one. Basil is an ass, but Sybyl doesn’t actually work, Manuel is pleasant, but pretty incompetent. And Polly isn’t really the voice of reason. I guess it is an entire show written around being annoying. So it can work.
Oh hell yeah.
David on LA Law. People people people!
btw, it’s “Just Shoot Me”, an admittedly lame play-on-words re: camera=gun.
Actually, that show was the only place where I did NOT find David Spade annoying.
But back to my title question - seems like several posters talking about A) characters where their annoyingness ADDS to the show. Others are talking about B) characters that are not supposed to be annoying but are, and still others are talking about C) characters whose annoyingness is supposed to add to the show but they are just annoying. And frankly some posts I am not sure which of the 3 the poster thinks the character is in.
My C) list would be huge. A lot of people love the deliberately annoying characters, but they usually just annoy me. There are a few exceptions, e.g., I did find Karen Walker funny on Will&Grace, at least for a while, but after a couple seasons I ended up hating everyone on that show.
My most annoying TV characfter of all time is Gary Coleman on Diff’rent Strokes. God I hated that show. I even hate that they spelled it “Diff’rent”.
The Professor in Gilligan’s Island
For unintentionally annoying, I’d have to go with Rachael on Friends. She was such a whiner.
I would draw a line between characters that are meant to be irritating to other characters, but that I find entertaining – Frank Burns, Michael Scott, Sheldon Cooper and Ted Baxter, e.g. – and characters that I personally find irritating.
An example of the latter, on a show I actually liked, was Mary (Nikki Cox) on Las Vegas. Insufferably whiny and needy. All the other characters on the show loved her, and I couldn’t stand her.