In 1910 The Portland Beavers won their second Pacific Coast League title, achieving a remarkable record in the process; the Beavers pitchers threw a baseball record 88 consecutive shutout innings.
Brandon Webb in 2007 with 42 1/3 consecutive scoreless innings.
Brad Ziegler Oakland Athletics 2008 32 innings still going
The Red Sox record is 45 shutout innings by Cy Young of the 1904 team. During this streak he pitched 25 1/3 hitless innings, still a major league record. The American League record for shutout innings is 56 by Walter Johnson of the 1913 Washington Senators.
Cy Young had 25.1 consecutive no-hit innings in 1904 for the red sox.
Babe Ruth with 29 2/3 consecutive scoreless innings pitched in World Series play.
The record is 59 by Orel Hershiser of the Los Angeles Dodgers between August 30, 1988 and September 28, 1988. Don Drysdale did it back in either the 50's or the 60's, but he had a record 58.2 scoreless innings, which basically means 6 straight shutouts plus 2.2 extra innings. Orel Herseiser broke that record back in 1988 with 59.0 consecutive scoreless innings
The Major League Baseball team with the most consecutive scoreless innings is 56 by the Pittsburgh Pirates. They have held that record since 1903.
Cal Ripken Jr. played 8243 consecutive innings between June 5, 1982 and September 14, 1987. That is believed to be the record but is unofficial since MLB does not recognize consecutive innings played as a statistic.