It is important to note that your church may not allow any individual to use church assets to further their private business without fair compensation to the church. In practice, the church could rent the organ/piano to the individual and allow that person to give lessons using church assets.
Essentially, as long as the rent charged equals fair market value, then the church can authorize the private use of church assets. However, this arrangement will generate unrelated business income and may cause the church to lose its property tax exemption on the rented asset. I do not recommend this option.
To learn more, see the downloadable resource Understanding Unrelated Business Income.