Is it possible to lose your voice due to allergies?

Hoarseness. Chronic post-nasal drainage can irritate the vocal cords and lead to hoarseness. As can laryngopharyngeal reflux and gastroesophageal reflux. Just a few of many causes.
Yes. Yes. This is a common complaint of patients with chronic allergies. Copious post nasal drainage, inflammation in throat and larynx, excessive mucous congestion in throat can cause you to lose your voice.