The priest is among those who goes to Christ not only in the sacrament of Holy baptism, but also is a direct servant of God. Engagement ring, symbolizing the connection between two people starred in a sign that the priest in the first turn, connected directly with God. In addition, the priest is not only a direct servant of the Lord, but also people wanting to find their way to God. When the wife of a priest has the full right to wear the ring, because she was not ordained.
In addition, there is a practical reason. The priest is the celebrant of the sacrament of the Eucharist (communion). That's when the prayer of the priest the grace of the Holy spirit descends on pre-prepared bread and wine. This divine grace, and contributes to the fact that the bread and wine become the Body and Blood of Christ. Thus, in the process of preparation of the substance for the sacrament, the priest must ensure that no crumbs of bread, and then the Body of Christ has not gone anywhere. In order to avoid situations when part of the Body of the Lord can get under the ring, wedding symbol is removed. Not a single crumb of the consecrated Holy gifts should not be lost. This reflects the reverence of the priest before the greatest Shrine of Orthodoxy.
Thus, it appears that the priests do not wear wedding rings as a sign of total dedication to God and for the sake of reverence for the Holy Body and Blood of Christ.