I can only speak for North American professional sports and that would be the Boston Bruins of the NHL who made the playoffs for 29 consecutive seasons from 1967-68 through 1995-96.
The Boston Celtics
The Washington Nationals / Montreal Expos franchise has the longest current streak of seasons not making the playoffs. They have not made the playoffs since 1981.
Through the 2008 season, the Steelers longest streak of seasons in the playoffs is 8 (1972-1979). They had a 6 season streak between 1992-1997.
Right now it's the Mariners. Until this year, the longest streak was held by the Royals and the second-longest the Blue Jays...but the Royals are in the Series this year and it is what it is.