In the World Series, does the American League team have a disadvantage by not being able to use a designated hitter at the National League club's home ballpark? 

The DH isn't the only concern to deal with either. AL pitchers have to hit—or more likely, bunt—something only required of them in nine games during the regular season. Managers have to use their bench more frequently, especially when a pitcher comes out of the game and his spot in the batting order comes up.