I am a professional musician and recently accepted an organist/pianist position with a new church. I have been told by the finance person that allowing me to teach lessons to church members 2 afternoons a week in a room of the church would jeopardize the church's 501(c)(3) status. I have offered to pay rent, or fee, etc., but she stands negative on this matter. Is she correct?
No. The incidental use of the church facilities for such purposes would not jeopardize the federal tax-exempt status so long as the organization carries on a legitimate program of religious activities. At the worst, the net income might be taxable as unrelated business taxable income. But rental income is not taxable if the property is not mortgaged. Even if the building is mortgaged, or if your payment is treated as a licensing fee and not rent, it might be considered related income if the program is structured as an educational opportunity for members of the church.
A charity always needs to consider whether use by others might affect its local real estate tax exemption, and that is a matter of state law varying in each state. But, particularly if the program is limited to members of the church, it seems unlikely that a use such as you propose would jeopardize an exemption.