Anyone know the record for consecutive innings by starters in baseball?

In 1968 the Detroit Tigers starters threw 12 complete games in a row. So that's 108 innings right there. I can't find much on the early 1900s but I'm pretty certain several teams from that time probably have streaks over 25 games in a row.

Since 1956 the Tigers, as Ben mentioned, hold the record with 12 complete games in a row. They went 10-2 during that span.

In 1980, Oakland starters threw 9 consecutive CGs, going 6-3, and the 1956 White Sox had 8 in a row (5-3).

Since 1997, four teams have had 3 consecutive complete games: '98 Braves, '99 Twins, '00 A's, '04 Twins.

I didn't find a listing for consecutive innings thrown by a team's starting pitchers.

I don't know if this is (part of) the record, but the 1881 Troy Trojans are worth knowing about. Despite the unfamiliar name, this was a National League team.

The entire pitching staff consisted of Tim Keefe and Mickey Welch, both now members of the Hall of Fame and the 300 Wins Club.

In 1881, the Trojans played 85 games. Keefe started 45, Welch started 40 -- and they threw a complete game each and every time. The records show they posted 771 innings. So depending upon how the 1880 season (81 CG in 83 games played) concluded and the 1882 season (81 of 85) began, the number of innings is ever larger.