Hal Holbrook probably has the record for this; he started performing as Mark Twain (originally in a play with his then-wife playing several female characters) since 1954 and is still playing the role in his one-man show in 2012. However, since Twain (or Clemens-as-Twain anyway) is a historical figure rather than fictional character and since it’s rarely been filmed (officially once, in the 1960s, with a few bootlegs) it’s perhaps not as impressive.
Leonard Nimoy first played Spock in 1966, last played him on film in 2009, and did a voiceover as Spock on an episode of *Big Bang Theory *this year, so he has about a 46 year run. Anyone on film or TV who can beat or match that in playing the same character.
Susan Lucci as Erica Kane on “All My Children” - 1970-2011
William Roache has played the character Ken Barlow in the UK soap opera Coronation Street continuously since it’s debut in 1960. From a quick wiki, it’s always been broadcast at a frequency of at least twice a week, (and currently is five episodes a week) so I’m guessing he’s made the most appearances as the same character.
Frances Reid played Alice Horton on Days of our Lives from the show’s debut on November 8, 1965 until December 26, 2007, and made occasion appearances until her death in 2010.
James Arness (1923–2011) as Matt Dillon needs to be considered, I should think.
Clearly someone needs to get James Cameron to make this movie.
He probably wins for time period as the same character on the same show, but he’s not even close for number of appearances. According to his IMDB page, he’s appeared in 1360 episodes of Coronation Street.
There are quite a few soap operas that have run, for years, at the rate of four or five episodes a week.
Deidre Hall has appeared in 2002 episodes of Days of Our Lives (1976-2012).
Ray Meagher has been in 2206 episodes of the Australian show Home and Away (1988-2012).
Kate Linder has been in 2580 episodes of The Young and the Restless (1982-2012).
Eileen Fulton played the much married Lisa on As the World Turns for 50 years (1960-2010) minus two breaks.
Harpo Marx seems to have played his mute, fright-wigged harpist as early as 1911 and appeared as that character as late as 1962, a total of 51 years.
Ah right. Looks like he had a few big gaps in the 70s and 80s. Impressively, he seems to be on the show most of the time for the past couple of years. Even more impressively, someone is documenting this on IMDB 
I’ve convincingly played myself for 35 years, three months, two weeks and a day.
And counting.
Moe Howard played the “Moe” character of the 3 Stooges from 1930 (or so) to just before his death in the early 70’s, for over 40 years, almost 50 depending on when the character of Moe became established.
Desmond Llewellyn played Q in the James Bond movies for 36 years.
Do the folks on Sesame Street count? Even aside from the muppeteers, Emilio Delgado has been playing Luis since '71, and Bob’s been doing his thing since the '60s…
Majel Barrett-Roddenberry was in Trek as an actress and the computer voice for about 44 years.
Don Hastings was on As the World Turns for 50 years, playing Dr. Robert Hughes from 1960 until the show went off the air in 2010. Except for a couple of six-month breaks, he was a member of a TV show cast continuously from 1948 (starting with Captain Video), though as different characters until he became Hughes.
Larry Fine was active almost as long as Moe. In fact, he joined with Ted Healy as one of his stooges in 1925, along with brothers Moe and Shemp Howard. Larry remained active as a Stooge until he suffered a stroke in 1970, for a total of 45 years.
Your mother told me me you’re not actually very convincing. 
Sesame Street premiered November 10, 1969.
Yes, about that. The producers and I would like to see you backstage. Do you have a moment?
It’s not the record, but Fred Rogers “played” Mr. Rogers for 43 years (1968-2001).