http://iscoresports.com/

When I change to a new pitcher, he assumes all of the old pitcher's stats. It is as if the starting pitcher never played
I think that I am doing the substitution correctly.
I click the pitcher's name on the iScore display of the field. A list of the players pops up. I select my new pitcher. However, the new pitcher assumes all of the old stats, just like he has been pitching the entire game.

Am I doing something wrong or is this a bug?

Your description of how to do it sounds like you are doing it correctly, however your description of the results sounds like you are doing something different.

Tapping the position and changing players is the proper way to make a substitution.
Using Misc->Show starting lineup is the method used to change the player that has played in that position from beginning of game (the results you are seeing).

Are you certain you did not do this?

Here is a link to how to make substitutions:
http://iscoreleagues.com/iscorewebsite/ ... lineup.php" onclick="window.open(this.href);return false;

I've gotten tripped up by this twice this season by forgetting to click the Save button after selecting the new pitcher.

Your milage may vary...

When you say you click on the pitchers name, are you clicking the pitcher position on the fielding illustration, or are you clicking the pitcher in the lineup section?

Tap the pitcher position in the field, then iScore will ask you to tap who the pitcher is and provide a list of all players on the team. Select the correct pitcher. That is a substitution.