The question this raises, then, is whether the injury would have affected Anthony in any meaningful way, and the answer there is no. If the news leaked as an excuse for Anthony's exit from the playoffs, it's not a very good excuse.
The injury, thought to be a labrum tear, is one seen more often in baseball, but it is not unheard of in basketball. Dwight Howard of the Los Angeles Lakers also played through a similar tear over the latter half of the season.
As with Howard, maintaining the shoulder and keeping it functional would be the key factor. The medical staff would have several options for working on things, up to and including painkilling injections.
Injections are, however, an unlikely choice for treatment. Even with the injured shoulder being on his non-dominant side, not being able to feel the shoulder is likely to have a negative effect on the mechanics of his shot. And really, any sort of normal, uncontested motion (like a jump shot) should not have caused any pain for Anthony.
It is instead other motions—those by which Anthony is not often troubled—that would be the biggest issue.
For Howard, the problem with his shoulder was exacerbated when the arm was pulled down and away from his body as he was reaching for rebounds or blocking shots. Those are two motions on which Anthony's game doesn't depend.
While that type of motion could occur in a normal jump shot, it's not normally the way a shot is contested. Fouls occur when the hands or elbow are hit or the body is jostled. The hands are seldom pulled away or back from the shot.
Additionally, the other mechanisms that cause pain due to a labrum tear do not appear to be in effect. Torn labrums tend to cause pain on deceleration, more of a concern for overhead athletes like pitchers or tennis players. There is no motion in a jump shot like a baseball pitch or a tennis serve.
Also, labral tears can cause a "catch" or clicking on motion. The shoulder doesn't move normally due to the inflammation.
In this case, Anthony would have been slow to get the ball up. That would normally cause an altered, lower arc or even a block, but neither of those things were especially noticeable during the playoffs.
Now that his season is finished, Anthony will take the next month or so to see whether rest alone will help clear up the pain and inflammation. If surgery is necessary, it is a simple arthroscopic procedure to clean up the damage inside; it's very unlikely to require more invasive techniques like re-anchoring the labrum.
Anthony would, in all likelihood, be ready for training camp.
In most situations like this, where an injury is known during the series, it comes off as nothing more than an excuse to have it "leak" like this afterward. Isola's poor relationship with the Knicks suggests that the leaks could have come from Anthony's camp.
Speculation aside, Anthony played well, leading his New York Knicks to the second round of the NBA playoffs despite a shoulder injury and showing no discernible effects. He shouldn't need any excuses.