There's hundreds of hours of video essays and reviews that discuss the story of Bioshock Infinite to death, so I'll just add some ideas that I haven't seen in other reviews.
On the surface, Bioshock Infinite presents Columbia as a White, Christian dystopia that brutally mistreats Black and Irish labourers. Columbia has futuristic technology that allows its buildings to defy gravity and fly in the air, yet instead of this technology improving the lives of the underclass, the labourers are little better off than slaves.
Black and Irish workers are shown in a floating anti-gravity platform doing menial labour, breaking rocks with hammers. Rocks that were presumably lifted from the ground, and which can presumably be broken by dropping them back down to the ground. Therefore that audience concludes that the state of affairs cannot have any economic excuse, but is a result of racial malice.
To me, the game strongly parallels 20th century sociology, in particular The Authoritarian Personality by T.W. Adorno.
Adorno, drawing on ideas from Freud, explains World War II by placing Christianity, nationalism, and (in the extreme) fascism as a form of psychosis. This is contrasted with American liberal democracy, which is the product of a stable and healthy psychology.
People have criticized Adorno for his methodology and bias, but I don't think any of that is too radical a perspective on WWII, and I don't think many people would defend Columbia and Father Comstock as being psychologically healthy.
The ending is where things take a darker turn... Spoilers ahead:
The revelation that Booker DeWitt is Father Comstock from another reality signifies that DeWitt carries the seed of The Authoritarian Personality. In another world, Elizabeth, his daughter, brings World War III to New York.
DeWitt is Dutch for "The White" (yourself, the gamer)
It's not enough for White People to embrace liberal democracy. It's not enough for America to defeat Nazi Germany.
"The White" will always contain the potential to become another Hitler. Therefore to rid the world of Hitler, "The White" must be exterminated in every parallel reality.
...what the hell?