The Bible addresses the celibacy of church leaders, but not celibacy of priests. In regards to celibacy of church leaders, in 1 Corinthians chapter 7
“But I would have you without carefulness. He that is unmarried careth for the things that belong to the Lord, how he may please the Lord: But he that is married careth for the things that are of the world, how he may please his wife. There is difference also between a wife and a virgin. The unmarried woman careth for the things of the Lord, that she may be holy both in body and in spirit: but she that is married careth for the things of the world, how she may please her husband.” 1 Corinthians 7:32-34
In some instances, celibacy has a positive impact on ministry. If a church leader is free from spousal and familial responsibilities, he can better focus on ministering to others. Jesus mentions some becoming “eunuchs” for the kingdom of God.
“For there are some eunuchs, which were so born from their mother's womb: and there are some eunuchs, which were made eunuchs of men: and there be eunuchs, which have made themselves eunuchs for the kingdom of heaven's sake. He that is able to receive it, let him receive it.” Matthew 19:12
Celibacy is definitely allowed for church leaders, and to a certain degree, it is encouraged. However, Scripture nowhere requires celibacy for those serving in positions of church leadership.
In 1 Timothy 3:1-13 and Titus 1:6-9, the Apostle Paul assumes that elders, bishops, overseers, and deacons will be married. Notice the phrases “the husband of one wife” (1 Timothy 3:2, 12; Titus 1:6), “he must manage his own family well” (1 Timothy 3:4,12), and “his children obey him with proper respect” (1 Timothy 3:4; Titus 1:6). While these Scriptures are not a requirement for church leaders to be married, they most definitely present an allowance for church leaders to be married. It is therefore anti-biblical for any church to require celibacy of its leaders.
Why, then, does the Roman Catholic Church (and a few other Christian denominations) require celibacy of priests /church leaders? The celibacy of priests has an interesting history. The first official church statements requiring celibacy appeared at the Councils of Elvira (A.D. 306) and Carthage (A.D. 390), although clerical celibacy, to a lesser degree, definitely predated these councils.
Ultimately, though, celibacy became the official requirement of the Roman Catholic Church due to the practice of nepotism. Church leaders were giving their children positions in the church, despite a lack of any qualifications or training. Further, church leaders were giving church property to their descendants. As a result, the Roman Catholic Church mandated celibacy in order to keep its priests from having familial attachments which made nepotism attractive.
Again, the Bible encourages, but does not demand celibacy of priests / church leaders. In fact, Paul recognizes that most church leaders will be married. The Roman Catholic requirement of celibacy is a sad example of the Church taking something that the Bible encourages and transforming it into a requirement in order to protect its own interests. Sadder still is the damage that has been done as a result of the Roman Catholic Church’s anti-biblical requirement. Men whom God has not gifted or called to be celibate (1 Corinthians 7:7) are being required to be celibate, and the result is tremendous failures in the areas of adultery, fornication, and the sexual abuse of children.