What is your definition of "significant"? Is that to say X musical was doing average business before it won a Tony for Best Musical and became a run away hit?
MEMPHIS wasn't exactly a hot ticket before it's win, but they did strong business after and have been enjoying a profitable run (profitable meaning not loosing money each week). I would say the Tony has helped it's box office significantly. BILLY ELLIOTT was one of the most popular shows in town prior to the Tony win. If it didn't win the Tony, would it still be making close to $1m each week? I didn't really follow grosses prior to the 2008/2009 season, so I can't speak about previous years.
"If this is going to be a Christian nation that doesn't help the poor, either we have to pretend that Jesus was just as selfish as we are, or we've got to acknowledge that He commanded us to love the poor and serve the needy without condition and then admit that we just don't want to do it." -Stephen Colbert