Many pitchers would continue to have their careers ended by batting and baserunning related injuries. Several sluggers will retire earlier than OTL. This goes beyond guys who became full-time DHs, many teams have used the DH as a way to rest veteran players.
Without the massive success of the DH, the league may be less inclined to innovate in the future. You could see an acceleration of baseball's OTL decline as the sport continues to struggle to win over Generation X audiences and adapt to the TV era.
Overall this would have big butterflies for the game, but the details are impossible to predict given the fickle nature of the sport.