At the high school age and frequency of games it shouldn't hurt you to run after a game. During a college season, I wouldn't advise it, and after a pro game you are just asking for injuries.
I agree that a bike would be a better choice.
As for the coach who makes the catchers run when the pitchers walk guys, i like it. I hated it when I had to do the same in college, but it really taught me a lot. For example, i didn't set up on the black with the count 0-0, 1-0, 1-1, 2-2, or 3-2. Turns out hitters get themselves out more often then not, and pitchers give them too much credit by nibbling at the corners. In addition, it also forced me to get on my pitchers. Forced me to tell them the way it is and be tougher.