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.
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.
Through the 2008 season, that is the Washington Nationals/Montreal Expos franchise whose last (and only) playoff appearance was in 1981.
The Buffalo Bills were last in the playoffs during the 1999 season.
The Steelers were established in 1933. In 1947, they made their first playoff appearance. They did not make the playoffs again until 1972. The longest the Steelers missed making the playoffs since that time, was a four year span (1985-1988).
Montreal has been playing since 1917, making them the longest team to play consecutive seasons.