The Boston Celtics
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 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.
Deleted previous one. Lakers did not make the playoffs in the 2004-2005 season. The real answer is the San Antonio Spurs, who have made the playoffs every year since 1998.