Dryden falls well short at 12 games, though I'm guessing that would be good enough for second on the list.
Depending on interpretation, Smith probably gets credit for 20 consecutive wins. According to HSP, Chico Resch lost three of the four games in the 1979 semi-final loss to the Rangers, so I don't think you can credit Smith (went 2-1) with losing the series. In the quarter-final series, Smith and Resch each got credit for two wins. The 1977 semi-final saw Smith credited with the loss in three of four against Montreal. That appears to be the last series he was definitively the losing goalie for until the 1984 final. So if he gets credit for 1979 QF, it's 20 straight wins.
I wonder what the record is for most consecutive wins from the start of a career? Niemi must be close by now.