Hypothetically, if you had an Atheist looking to make money by becoming a priest, would the church be legally allowed to deny him the job because of his religion (or lack of), assuming he's still willing to talk about God as if he is Christian? I am to understand that you cannot deny someone a job on the grounds of religious beliefs, I'm wondering if it works the same way for religious figures in churches. (This can be applied in many other ways, a Muslim looking to become a Christian priest, etc.)
Yes: It is legal to deny someone a job as a priest because he is an atheist.
Churches are allowed to discriminate in employment based upon religion. See, for example, the EEOC compliance manual. This says, in the pertinent part (citations included after the quoted material):
Under Title VII, religious organizations are permitted to give employment preference to members of their own religion. The exception applies only to those institutions whose “purpose and character are primarily religious.” That determination is to be based on “[a]ll significant religious and secular characteristics.” Although no one factor is dispositive, significant factors to consider that would indicate whether an entity is religious include:
- Do its articles of incorporation state a religious purpose?
- Are its day-to-day operations religious (e.g., are the services the entity performs, the product it produces, or the educational curriculum it provides directed toward propagation of the religion)?
- Is it not-for-profit?
- Is it affiliated with or supported by a church or other religious organization? 
This exception is not limited to religious activities of the organization. However, it only allows religious organizations to prefer to employ individuals who share their religion. The exception does not allow religious organizations otherwise to discriminate in employment on protected bases other than religion, such as race, color, national origin, sex, age, or disability. Thus, a religious organization is not permitted to engage in racially discriminatory hiring by asserting that a tenet of its religious beliefs is not associating with people of other races. Similarly, a religious organization is not permitted to deny fringe benefits to married women but not to married men by asserting a religiously based view that only men can be the head of a household.
Sex Discrimination Not Excused
Justina works at Tots Day Care Center. Tots is run by a religious organization that believes that, while women may work outside of the home if they are single or have their husband’s permission, men should be the heads of their households and the primary providers for their families. Believing that men shoulder a greater financial responsibility than women, the organization pays female teachers less than male teachers. The organization’s practice of unequal pay based on sex constitutes unlawful discrimination.
Courts have held, based on First Amendment constitutional considerations, that clergy members cannot bring claims under the federal employment discrimination laws, including Title VII, the Age Discrimination in Employment Act, the Equal Pay Act, and the Americans with Disabilities Act, because “[t]he relationship between an organized church and its ministers is its lifeblood.” This “ministerial exception” comes not from the text of the statutes, but from the First Amendment principle that governmental regulation of church administration, including the appointment of clergy, impedes the free exercise of religion and constitutes impermissible government entanglement with church authority. Thus, courts will not ordinarily consider whether a church’s employment decision concerning one of its ministers was based on discriminatory grounds, although some courts have allowed ministers to bring sexual harassment claims.
The ministerial exception applies only to those employees who perform essentially religious functions, namely those whose primary duties consist of engaging in church governance, supervising a religious order, or conducting religious ritual, worship, or instruction. The exception is not limited to ordained clergy, and has been applied by courts to others involved in clergy-like roles who conduct services or provide pastoral counseling. However, the exception does not necessarily apply to everyone with a title typically conferred upon clergy (e.g., minister). In short, in each case it is necessary to make a factual determination of whether the function of the position is one to which the exception applies.
The relevant footnotes:
 Section 702(a) of Title VII, 42 U.S.C. § 2000e-1(a), provides:
This subchapter shall not apply to . . . a religious corporation, association, educational institution, or society with respect to the employment of individuals of a particular religion to perform work connected with the carrying on by such corporation, association, educational institution, or society of its activities.
Section 703(e)(2) of Title VII, 42 U.S.C. § 2000e-2(e)(2) provides:
it shall not be an unlawful employment practice for a school, college, university, or educational institution or institution of learning to hire and employ employees of a particular religion if such school, college, university, or other educational institution or institution of learning is, in whole or in substantial part, owned, supported, controlled, or managed by a particular religion or by a particular religious corporation, association, or society, or if the curriculum of such school, college, university, or other educational institution or institution of learning is directed toward the propagation of a particular religion.
While Congress did not include a definition of the § 702(a) term “religious corporation” in Title VII, at least one judge has argued that the legislative history indicates that Congress intended “the § 703(e)(2) exemption to require a lesser degree of association between an entity and a religious sect than what would be required under § 702(a).” See LeBoon v. Lancaster Jewish Cmty. Ctr., 503 F.3d 217, 237 (3d Cir. 2007) (Rendell, J., dissenting).
Executive Order 13279, Equal Protection of the Laws for Faith-Based and Community Organizations, issued on December 12, 2002, provides that certain faith-based organizations that provide social programs can deliver those services and make hiring decisions on the basis of their religious beliefs even if they receive federal funding. See 67 Fed. Reg. 77,141 (12/16/02). The Guidance to Faith-Based and Community Organizations on Partnering with the Federal Government, http://www.whitehouse.gov/government/fbci/guidance_document_01-06.pdf (last visited July 2, 2008), issued by the White House Office of Faith Based and Community Initiatives, explains that while religious organizations that receive federal funds to provide social services may choose to hire persons of the same religion, they are also subject to federal, state, and local employment and anti-discrimination laws, such as Title VII.
 Townley, 859 F.2d at 618; accord Hall v. Baptist Mem. Health Care Corp., 215 F.3d 618, 624-25 (6th Cir. 2000) (college of health sciences qualified as a religious institution under Title VII because it was an affiliated institution of a church-affiliated hospital, had direct relationship with the Baptist church, and the college atmosphere was permeated with religious overtones).
 Townley, 859 F.2d at 618; see also Killinger v. Samford Univ., 113 F.3d 196 (11th Cir. 1997) (Baptist university was “religious educational institution” where largest single source of funding was state Baptist Convention, all university trustees were Baptists, university reported financially to Convention and to Baptist State Board of Missions, university was member of Association of Baptist Colleges and Schools, university charter designated its chief purpose as “the promotion of the Christian Religion throughout the world by maintaining and operating institutions dedicated to the development of Christian character in high scholastic standing,” and both Internal Revenue Service (IRS) and Department of Education recognized university as religious educational institution).
 Townley, 859 F.2d at 619 (manufacturer of mining equipment, whose owners asserted that they made a covenant with God that their business “would be a Christian, faith‑operated business,” is not a religious organization because it is for profit; it produces mining equipment, a secular product; it is not affiliated with or supported by a church; and its articles of incorporation do not mention any religious purpose). Cf. EEOC v. Kamehameha Sch./Bishop Estate, 990 F.2d 458, 461 (9th Cir. 1993) (non-profit school not “religious” for Title VII purposes where ownership and affiliation, purpose, faculty, student body, student activities, and curriculum of the schools are either essentially secular, or neutral as far as religion is concerned).
 See Corp. of the Presiding Bishop of the Church of Jesus Christ of Latter-Day Saints v. Amos, 483 U.S. 327 (1987) (a nonprofit church-run business does not violate Title VII if it refuses to hire anyone other than members of its own religion, even for enterprises or jobs that are not religious in nature).
 Killinger, 113 F.3d at 200 (School of Divinity need not employ professor who did not adhere to the theology advanced by its leadership); Tirpanlis v. Unification Theological Seminary, 2001 WL 64739 (S.D.N.Y. Jan. 24, 2001) (seminary operated by Unification Church cannot be sued for religious discrimination by Greek Orthodox employee who was allegedly terminated for refusing to accept the teachings of the Unification Church).
 Ziv v. Valley Beth Shalom, 156 F.3d 1242 (Table), 1998 WL 482832 (9th Cir. Aug. 11, 1998) (unpublished) (religious organization can be held liable for retaliation and national origin discrimination); DeMarco v. Holy Cross High Sch., 4 F.3d 166 (2d Cir. 1993) (religious institutions may not engage in age discrimination).
 EEOC v. Fremont Christian Sch., 781 F.2d 1362 (9th Cir. 1986) (religious school violated Title VII and the Equal Pay Act when it provided “head of household” health insurance benefits only to single persons and married men).
 McClure v. Salvation Army, 460 F.2d 553, 558-60 (5th Cir. 1972); see also Hollins v. Methodist Healthcare, Inc., 474 F.3d 223 (6th Cir. 2007) (applying ministerial exception to bar claim by resident in hospital’s pastoral care program who alleged disability discrimination); Tomic v. Catholic Diocese of Peoria, 442 F.3d 1036 (7th Cir. 2006) (applying ministerial exception to bar age discrimination claim brought by Catholic Diocese music director who was terminated following a dispute with the bishop’s assistant regarding what to play during the Easter Mass); Hankins v. Lyght, 441 F.3d 96 (2d Cir. 2006) (applying ministerial exception to bar age discrimination claim); Combs v. Central Texas Annual Conf. of United Methodist Church, 173 F.3d 343 (5th Cir. 1999) (barring claim because court could not determine whether an employment decision concerning a minister was based on legitimate or illegitimate grounds without entering the constitutionally impermissible realm of internal church management); EEOC v. Catholic Univ. of America, 83 F.3d 455 (D.C. Cir. 1996) (ministerial exception barred Title VII sex discrimination claim brought by tenured member of Catholic University’s department of religious canon law); DeMarco v. Holy Cross High School, 4 F.3d 166 (2d Cir. 1993) (ministerial exception inapplicable to parochial school teacher’s age discrimination claim because employer’s contention that teacher was terminated specifically for failing to attend Mass and to lead his students in prayers could be evaluated without risk of excessive entanglement between government and religious institution); Guianan v. Roman Catholic Archdiocese of Indianapolis, 42 F. Supp. 2d 849 (S.D. Ind. 1998) (ministerial exception inapplicable to parochial school teacher’s age discrimination claim, even though teacher taught at least one class in religion per term, and organized one worship service per month, since vast majority of teacher’s duties involved teaching math, science, and other secular courses).
 Rayburn v. Gen. Conference of Seventh‑Day Adventists, 772 F.2d 1164, 1169 (4th Cir. 1985).
 Rweyemamu v. Cote, 520 F.3d 198 (2d Cir. 2008) (Title VII race discrimination claim by African-American Catholic priest challenging denial of promotion and subsequent termination was barred by the ministerial exception); Petruska v. Gannon Univ., 462 F.3d 294 (3d Cir. 2006) (ministerial exception bars Title VII sex discrimination claim by female Catholic chaplain against school, alleging that she was forced out as chaplain after she advocated on behalf of alleged victims of sexual harassment and spoke out against the school’s president regarding alleged sexual harassment and discrimination against female employees); Werft v. Desert Southwest Annual Conf. of the United Methodist Church, 377 F.3d 1099 (9th Cir. 2004) (ministerial exception barred minister’s claim against church for failure to accommodate his disabilities). However, some courts have ruled that the ministerial exception does not bar harassment claims by ministers, but rather only applies to claims involving matters such as hiring, promotion, and termination. See Elvig v. Calvin Presbyterian Church, 375 F.3d 951 (9th Cir. 2004) (ministerial exception does not bar sexual harassment claim by minister), reh’g denied, 397 F.3d 790 (9th Cir. 2005) (two concurring and three dissenting opinions); Bollard v. California Province of the Soc’y of Jesus, 196 F.3d 940 (9th Cir. 1999) (novice’s sexual harassment claim could be maintained without excessive entanglement between church and state because religious order did not offer a religious justification for the alleged harassment, and plaintiff did not seek reinstatement or other equitable relief); Dolquist v. Heartland Presbytery, 342 F. Supp. 2d 996 (D. Kan. 2004) (First Amendment Establishment and Free Exercise Clauses did not preclude minister from pursuing Title VII sexual harassment claim against her church, because claims did not involve choice of clergy); see also Bryce v. Episcopal Church in the Diocese of Colorado, 289 F.3d 648, 657-59 (10th Cir. 2002) (although “employment decisions may be subject to Title VII scrutiny, where the decision does not involve the church’s spiritual functions,” minister’s Title VII harassment claim was subject to dismissal because it was based on communications protected by the First Amendment under the “church autonomy” doctrine; the doctrine is broader than the ministerial exception and bars civil court review of internal church disputes involving matters of doctrine and church governance).
 Geary v. Visitation of Blessed Virgin Mary Parish Sch., 7 F.3d 324 (3d Cir. 1993) (lay teacher at church‑operated elementary school not a minister); Dole v. Shenandoah Baptist Church, 899 F.2d 1389 (4th Cir. 1990) (lay teachers of private religious schools who “perform no sacerdotal functions [nor] serve as church governors [and] belong to no clearly delineated religious order” are not ministers despite their sincere belief that theirs is a ministry); but see EEOC v. Catholic Univ. of America, 83 F.3d 455 (D.C. Cir. 1996) (ministerial exception barred Title VII sex discrimination claim brought by tenured member of Catholic university’s department of religious canon law).
 Alicea‑Hernandez v. Catholic Bishop of Chicago, 320 F.3d 698 (7th Cir. 2003) (ministerial exception applied to Communications Director who was responsible for crafting the Church’s message to the Hispanic community); EEOC v. Roman Catholic Diocese of Raleigh,213 F.3d 795 (4th Cir. 2000) (ministerial exception applies to cathedral’s director of music ministry and part-time music teacher); Rayburn, 772 F.2d at 1168 (ministerial exception applies to associate pastor who had completed seminary training but was not ordained); Starkman v. Evans, 198 F.3d 173 (5th Cir. 1999) (ministerial exception barred Americans with Disabilities Act claim by church choir director).
 EEOC v. Southwestern Baptist Theological Seminary, 651 F.2d 277, 283 (5th Cir. 1981) (“[w]hile religious organizations may designate persons as ministers for their religious purposes free from any governmental interference, bestowal of such a designation does not control their extra‑religious legal status”).
This exemption has a constitutional dimension under the First Amendment free exercise clause so this rule cannot be changed, even by statute.
In the case of the atheist priest, I would argue that an atheist cannot actually do a priest's job. For example, I doubt that a Christian marriage would be valid if the priest was an atheist. It's like a steak house discriminating against a vegan chef who refuses to touch or cook any meat. That vegan chef just can't do the job in a steak house.
You might consider some other cases: Can a church owned hospital refuse to hire an atheist doctor? (Since being atheist makes you unable to be a priest in my opinion, but you can be a perfectly fine doctor). Or can a religious school refuse to hire an atheist religious educator? Or an atheist cleaner?