Are there any potential threats from bronchoconstriction?

Stop breathing. Bronchoconstriction, or narrowing of the airways due to swelling and/or spasm, decreases airflow into the lungs, and increases the effort needed to get that air in. This combination of stresses lowers the oxygen level, and if severe enough, a person can become weak, confused, collapse from exhaustion or even stop breathing altogether. This is how people can even die from an asthma attack.