I've seen it happen twice before, and both times it backfired.
Vancouver did it in Game #81 of the 2000 season against the Oilers because they needed a win to stay alive in the playoff race. Edmonton scored in the final minute of OT to clinch a playoff spot and knock Vancouver out.
Edmonton Oilers at Vancouver Canucks Box Score — April 7, 2000 | Hockey-Reference.com
The other time was a CBJ-LAK game late in the 2003 season. Same scenario as above, the Kings needed a win to stay alive. Columbus scored in the final minute of overtime.
Columbus Blue Jackets at Los Angeles Kings Box Score — March 25, 2003 | Hockey-Reference.com
ftr, if a team loses in OT because they pulled their goalie, they forfeit the loser point, and it counts as a regulation loss.
Edit: I think there was a team within the last few years that pulled the goalie OT because a shootout win didn't help them for tiebreaking purposes. I want to say it happened against the Avs and was in Colorado. The game ended in a shootout.