Would active noise cancellation increase or reduce the damage to ears? For example, in a hypothetical system with a constant level of external noise, would wearing noise-cancelling headphones increase or decrease the amount of damage to hearing from being in that environment? I feel like they should in theory decrease it by cancelling the waves but at the same time they're still adding sound to the environment to achieve their effect, so I'm not sure. So does wearing noise cancelling headphones damage your ears more or less than being in the same environment without them?
The answer to this question is felt to be a clear "reduce" in the other typical use case for noise-cancelling headphones - work in loud industrial sites. See for instance this description, which admittedly is biased source (a manufacturer of such headphones). I would strongly expect it to be the same in music.
The main issue is that damage to hearing comes from sustained exposure to overly loud sound levels. Passive and active noise reduction, when successful, reduces the loudness of such sounds, and so prevents/reduces damage.
I suppose that in instances where there are periodic unpredictable very loud sounds, which active noise cancellation may have difficulty fully controlling, against a backdrop of moderately loud sound, there could be a pathology where the moderately loud sound would be sufficiently annoying to make you stop before your hearing is (too) damaged, but the control of such sound would make you last longer enough that the partially-only controlled loud sounds while they insidiously damage your hearing. Sort of like crappy sunglasses can cause indirect harm by reducing your eyes'/brain's natural response to excess light and UV rays. But that's a secondary and -- on my part -- speculative effect which I would hope would normally be masked by the primary positive one.