So the tetany in breathwork is generally caused by the decreased CO2 concentration causing respiratory alkalosis (ie blood gets more alkaline and has a ph balance of > 7.5), which in turn causes the protein albumin to bind more strongly to calcium and not release it as it's supposed to, and calcium is an important regulator in voltage gated ion channels in neurons.

Long story short, your neurons get just a tad bit more excitable because calcium that usually acts like the bouncer to the hot club is busy snogging albumin. That has very little effect in places in the body, but in motor neurons that control your smallest muscles (face and hand), and in sensory neurons under your skin it does move the needle — that causes the muscles to contract and your skin to feel tingly, both exactly the same cause.

This is the reason people with epilepsy should _NOT_ do breathwork, but for otherwise healthy adults there are no negative long term effects of respiratory alkalosis — a few normal breaths to balance out your co2 and the symptoms will go away.