Is it biblical to have a board of trustees?
Governments exist by the authority of God (Romans 13:1-2). So long as the demands of a government do not violate the laws of God, there is every reason for Christians to be obedient to the government under which they exist.
When a congregation owns a building, the government requires that there be someone designated to own the land, whether it be one or more individuals or a corporation. The preferred method is a corporation because people come and go. The existence of a congregation generally exceeds the life of the people in the congregation. If individuals owned the property for the church, it would have to be bought and sold as the people came and went. If someone died while owning the property, it would pass through that person's inheritance. Such is both annoying and costly, especially if the heirs are not members of the church.
Having a corporation own the property is more expedient under most government laws because trustees can be named to fulfill the legal requirement, and they are easily changed as time passes. When people move or die, different people can be appointed to be the designated trustees.
The trustees have no authority in the church. They only exist to satisfy the government's need for someone to communicate with and act on behalf of the congregation in government matters (such as signing tax-exempt forms or setting up bank accounts). Typically, most congregations designate the elders and deacons as trustees of the congregation's property. If there are no elders and deacons, the men would have to be selected to act in that capacity for the congregation.
This need for representatives to communicate with the government does not change the organization of the church.