Like many others, when analyzing The Birdcage from a Queer perspective, I find the film problematic for several reasons. While one could argue that "all exposure" is good exposure, in The Birdcage, the audience is forced to encounter the relationship between Armand and Albert, a relationship that is ultimately constrained by the societal binary of male and female gender roles, and more specifically, the gender roles expressed within "the home."
One of the most clear depictions of this occurs when Armand and Albert's home must me temporarily renovated to appease their guests. With the intention of "de-gaying" the house, all of Albert's taste and decor are removed from the house, in exchange for more secular decoration. In this sense, the only way to expel Albert (who symbolizes the feminine nature of homosexual relationships) is to remove his material presence from the house. Albert's homosexuality, even in the eyes of his own partner was unsuitable for common society. In the film, the "remedy" to this problem was for Albert to assume a role on the complete opposite side of the spectrum, essentially masking any notion of homosexuality, solidifying the idea that a gender binary is the only way to succeed in the home.
Unfortunately, the problem that arises with this film is that stereotypes of LGBT relationships are not critically examined, but rather, reenforced by their presence in the film. By the very nature of their relationship (and notably during the Dinner Table scene) Albert and Armand present a homonormative expression of relationships between two men, where both men cannot express their own masculinity.
Thus, despite the notion of exposure at any cost, this film does little to improve stereotypes, and I believe that this is why it is not successful as an LGBT film. Where it is very entertaining and comical, and although it's intentions are most likely not negative, there is no progress as a result of this film.