That does not seem to be true. In fact, per the IRS, religions have fewer requirements for transparency than other non-profit groups.
"Annual Exempt Organization Return: Who Must File
Every organization exempt from federal income tax under Internal Revenue Code section 501(a) must file an annual information return
except:
A church, an interchurch organization of local units of a church, a convention or association of churches,
An integrated auxiliary of a church,
A church-affiliated organization that is exclusively engaged in managing funds or maintaining retirement programs,
A school below college level affiliated with a church or operated by a religious order,
Church-affiliated mission societies if more than half of their activities are conducted in, or are directed at persons in, foreign countries,
An exclusively religious activity of any religious order,
Tax Inquiries and Examinations of Churches
Congress has imposed special limitations, found in section 7611 of the Internal Revenue Code, on how and when the IRS may conduct civil tax inquiries and examinations of churches. The IRS may begin a church tax inquiry only if an appropriate high-level Treasury official reasonably believes, on the basis of facts and circumstances recorded in writing, that an organization claiming to be a church or convention or association of churches may not qualify for exemption, may be carrying on an unrelated trade or business (within the meaning of IRC § 513), may otherwise be engaged in taxable activities or may have entered into an IRC § 4958 excess benefit transaction with a disqualified person."
Tax Information for Churches and Religious Organizations