What are duties of the Trustee in the AME Church?

It's mainly a state matter.

Churches are usually recognized as an incorporated legal entity by the state. Often, the state requires that incorporated legal entities have trustees, which are the board that conducts the legal affairs of the church -- hold title to property, sign contracts, etc. So, many churches have opted to elect a board of trustees. Other churches will say that biblically there is no such church officer as trustee, so many churches believe that having trustees puts church organization out of alignment with scripture.

However, a church does not need to elect trustees to fulfill the state requirement. All it has to do is appoint someone (it could be an elder board) to act as trustees in the eyes of the state.

Some will argue that the elders have responsibility for spiritual matters and ministry of the word, so being a trustee is outside their area of expertise. No problem -- all the elders need to do is retain the role of trustee in the eyes of the state, but appoint members with expertise in buildings, property, etc, to advise them. They could even empower them with decision making.

Bottom line is that a state usually requires an incorporated entity to have trustees, so many churches formalize that in their bylaws.