The Yankees scored 12 runs in the first inning against the Orioles.
The most runs scored in an inning by both teams is 19 in the 8th inning of a game between Cleveland and Boston in 1977. Cleveland scored 13 runs and Boston scored 6. The most runs scored by a single team in aninning is 18 by Chicago against Detroit in the 7th inning of a game played in 1883.
That was the 1st and the 4th where 8 runs combined were scored in each inning.
In MLB history, the most runs scored by a single team in the ninth inning is 14 by the Baltimore Orioles against the Boston Beaneaters in 1894. The most runs scored by both teams in the ninth inning is 17 by the New York Giants and Boston Braves in 1912. The Giants scored 7 runs and the Braves scored 10 runs.
17 runs: Boston (17) vs. Detroit (0) on 6/18/1953
They scored 8 in the 2nd inning of some game. My source doesn't give the date, but that is the most they've scored in an inning this year.
The Cleveland Indians scored 14 runs in the top of the 2nd inning on April 18, 2009.
The most runs the Yankees have scored in one inning is 14in the 5th inning of a game against the Washington Nationals on July 6, 1920.
Brian Lara has scored 400* in an inning.