As per this previous answer to "Why do I feel small stings all over my body in stress?", hyperventilation is known to cause tingling, numbness, or muscle spasms in your hands, feet, and perioral regions. What is the mechanism for this effect?
The two proposed mechanisms for paresthesia/tetany secondary to hyperventilation are cerebral vasoconstriction and electrolyte imbalance.
Cerebral vasoconstriction: Cerebral blood flow decreases in a linear relationship to decreased PaCO2 (partial pressure of carbon dioxide). Thus, hyperventilation -> decreased PaCO2 -> decreased cerebral blood flow -> neurologic manifestations in distal extremities.
Electrolyte imbalance: Since CO2/HCO3- is a major pH buffer in our blood, hyperventilating removes H+ from our blood, leading to alkalosis. Serum H+ normally prevents calcium from binding to albumin in a competitive manner. Decreased serum H+ -> increased calcium binding to albumin -> local hypocalcemia, which can explain the paresthesia/tetany in distal extremities. Hypophosphatemia is also found in hyperventilation, which can contribute to tetany.
Source: Schwartzstein, RM. et al. Hyperventilation syndrome in adults. In: UpToDate (Accessed on September 14, 2020 from https://www.uptodate.com/contents/hyperventilation-syndrome-in-adults)