First of all: fundamentals... The primary duty of the priest is to act as a mediator between man and God. It follows that the most sacred duty of the priest is the recitation of the psalter, aka the Divine Office. Contrary to popular belief, the priesthood was not instituted to confer the sacraments and say Mass, these are tertiary functions. Therefore, a mute man could be a priest. Could he offer Mass? No. Could he forgive sins? That is a question I cannot answer. However, since a priest can hear confessions via a translator, it follows that he might be able to give absolution silently. By the same token, he might be able to give a blessing and benediction silently as well, merely willing it. But, at the very least, muteness would be an impediment. Whether dirigible, I do not know.
Our Lady of Fatima, pray for us, you are our only hope.