ideally no, just the opposite. the idea of ANC is that you detect the frequency and amplitude of incoming waves, then generate a second wave with the same amplitude, but 180 degrees out of phase (so that the trough of the second wave aligns with the peak of the first). a perfect implementation would totally "cancel" the original sound with destructive interference.
I'm not very familiar with consumer ANC headphones (I prefer passive attenuation), but I would expect the cancellation effect does not increase when you turn up the volume, as this would just be perceived as noise if it exceeds the amplitude of the ambient sounds. I guess ANC might encourage people to expose themselves to very loud noises that the device can't compensate for, but that's more user error than an inherent flaw in the technology. people routinely listen to music way too loud in quiet settings anyway.
* A "live" ANC process has no control over the environment from which it receives the audio signals that it acts adversarially against.
* When transmitting audio waves from one medium to another, there will be latency. Perhaps not much, but it will be there.
If you accept these two positions, then consider this:
* What happens when a sound wave that is being combatted (via phase inversion) suddenly stops, or inverts it's own phase? That's right, ANC could potentially double the amplitude of the frequency being combatted.
* I imagine that ANC technology takes advantage of latency to ensure that they don't damage people's hearing, but the nature of ANC requires low latency in general, otherwise you can't be sure that you are combatting the correct frequency (at which point you risk doubling the amplitude due to abrupt changes) - if someone more familiar with the actual algorithms could chime and correct me I will happily stand corrected :)
Second off, instantaneous sound is only a health issue when it's really loud, like a gunshot (130-140dB SPL) near the point at which the ear drum ruptures. That means that you need to be in an environment where the background noise is dangerously loud to begin with, and because of the way sound is made - this might be unlikely. Which is interesting, because early ANC did have these problems - when it was being used initially for military applications (helicopter/tank pilots iirc).
Lastly the important thing to remember is that ANC is usually part of a dual pronged approach to ear protection. Latency is a problem when you need to cancel high frequencies (where you get past about a quarter wavelength and interference can become constructive), but ANC excels at low frequencies (below about 500-1kHz it can be remarkable even). This is great because passive reduction strategies (sealing off the ear, thick padding, good fit/headband adjustment) are much more effective at high frequencies.
So TL;DR it was a problem, been fixed, and where it might happen is pretty rare for a consumer.
Also noise rarely spontaneously inverts phase at a particular frequency. That'd be weird.
