We are a 501(c)(3) church. A church member wants to offer an exercise program in our facility to members and to the community for a charge / profit. Does that affect our 501(c)(3) status even if we do not charge rent? Should we charge rent? The church does not have a mortgage.
It is highly unlikely that the incidental use of a church facility for an exercise class for members and others would affect your 501(c)(3) status as long as you continue a regular church program. Whether you charge rent for the space is up to you. For federal UBIT purposes, it would not be subject to unrelated business income tax (even if it is unrelated to your exempt purpose) so long as the building is not subject to acquisition indebtedness such as a mortgage.
Whether or not the activity would impact your local real estate tax exemption is a state law question. Exemption for churches is often based on a narrower consideration than the general exemption for charities. If this is going to be a significant activity, you might want to check with your local assessor.