See Mormon Film: Key Films of the Third Wave
This film marks the beginning of an era that views Mormonism more cynically and maturely, after a long spell of mostly positive depictions under the Production Code (most prominently "Brigham Young" and "Wagon Master"). It is also the first non-Mormon-made feature film to treat Mormonism in the present since perhaps "Married to a Mormon" in 1922.
In the book the Brigham Anderson character is not only identified as LDS, but as the son of an apostle. The film refrains from mentioning the Church, and in one scene where Anderson undresses to go to bed he is not wearing garments, but he is implied to be LDS by his being the only family depicted in the film, his thrice refusing three different items forbidden by the Word of Wisdom, and, most prominently, the moral pressure he feels from his unforgiving constituency.
Anderson's portion of the film is a subplot, though an important one, and the film was not seen as anti-Mormon or even relating to Mormonism; still, as mentioned, it marks the beginning of a trend that considers modern Mormons as flawed and sometimes sinning, thus preparing for more blatantly anti-Mormon films of the 70s through the present. It was, more specifically, the first film connecting Mormonism and homosexuality; the scene where Anderson enters the gay bar was the first depiction of such an establishment in the mainstream (i.e. MPAA accepted) American cinema--it is appropriate that it is a revolted Mormon character that took American film into this new era.
"Advise and Consent" is often described as the greatest film ever made on American politics (by Peter Bogdanovich, for instance). It was also Charles Laughton's last film. President Kennedy and other Washington elite were involved in its creation.