Anivia's ultimate is actually a 'sustained ability.' (It's the only one I can think of in the game.) This means you can still move and cast other abilities while it is up, but getting stunned or silenced will interrupt your ultimate.

This is different from toggled abilities which only turn off if you cast it again or if you run out of mana. (Someone silencing you just means you can't turn it off.)

I don't know how Zhonya's interacts with her ability, it's likely it will cancel it because it acts as a self-silence, but I haven't used it on her in ages (since it was Zhonya's Ring).