I think far more likely that pitchers simply won't be able to make it out of tough innings, without rest between pitches. That shouldn't add to injuries. It should reduce the number of pitches and probably number of IP many pitchers are capable of. If a pitcher is wise, he won't waste so many pitches trying to set up the hitter, especially after two strikes. Nola was guilty of that a lot in the playoffs. Fewer throws to 1B, more steals, BA already ticked up. I expect more runs, higher ERAs, but not more injuries, unless we have stubborn pitchers and managers who insist on sticking to their past pitches, IP, and not greatly reducing waste pitches. An increased premium on throwing a higher percentage of strikes, removing a starter who is in trouble, and a quality pen.
For the Phillies, it mainly impacts our aces, who need to make changes. They both clearly ran out of gas in last year's playoffs. This year's rule changes will only make things worse for them.