You're right, awards don't make a show stay open, money does. The thing is though, the awards influence the money. People listen to the critics and the awards. We've all seen it happen. We've all seen the closing notices that get posted within days after the Tony Award nominations/ceremony.
Same with reviews. I used to work for an off-broadway theatre. During previews, the house would be half full. Then the New York Times review would come out. If the review was good, the theatre would go from being half full to being sold out with a waiting list for cancellations practically overnight.