Apparently, there is no scientific evidence that ANC is or is not causing tinnitus.

ANC reduces background noise, which typically allows users to listen at lower volumes, thereby reducing total sound exposure to the ear. So if the user adapts their volume, that would lead to less risk of tinnitus. This works for me :)

But there are lots of people on forums suggesting that there is a link between tinnitus and ANC. One reason could be that ANC headphones allow you to listen very accurately to inner auditory signals, and if you already had some tinnitus, you might start to notice it.