April 21, 1934
According to Charlton’s Baseball Chronology, Moe Berg set a record for catchers by playing in his 117th consecutive errorless game.
Berg was a backup catcher for Washington from 1932 to the middle of 1934. He appeared in 141 games as a catcher with the Nats. B-R only seems to have season totals for errors, so it is not clear to me when his streak finally came to an end, though it was sometime between the April 21 game and July 20 of that year, the last game Berg played for Washington. He was released by the Nats and picked up shortly thereafter by Cleveland, where he finished the season. In all, Berg committed just one error with Washington.
The current AL record is held by a familiar player. Mike Redmond has not committed an error since 2005, the year he first joined the Twins. His streak stands at 232 games.