From interviews with director Christopher Nolan, it would appear that the goal was to create a non-specific setting (the film was actually shot in Pittsburgh and LA as well as New York and New Jersey), that felt both "operatic" and real.
From an interview with James Rocchi at MSN Entertainment:
MSN Movies: You knew you were doing a third film. Is the pressure to do things bigger, better -- is that from the studio? Is it from the
audience? Or is it that all from you? Are you the person saying,
"Let's do more with this"?
Christopher Nolan: Well, I was trying to push filmmaking from the point of view of the audience. I'm trying to make a film that I would
want to go see on a Friday night and lay down ten dollars or what have
you and be thrilled and moved by or whatever. And that's what was the
impulse, and so when you know you're dealing with characters the
audience have a relationship with, you're asking the audience to come
back to Gotham City. So you feel a huge creative responsibility to
give them a reason to come back to Gotham City.
From an interview with Emanuel Levy:
For the first time, three separate cities provided locations and
backdrops for Gotham City, with scenes filmed in Pittsburgh, Los
Angeles, and the city sometimes nicknamed Gotham, New York.
Nolan offers, “When you look back on the films, you can see the world
we’re living in reflected, but we don’t want to be specific about it.
We just come at the stories from the point of view of what concerns
us. What gives us fear? What gives us hope? What would require a hero
of Batman’s stature to rise up in our world?”
And from an interview with Alex Tucker at TV-Film-News:
**‘The Dark Knight Rises’ magnifies and resolves the main themes of the
trilogy, one of which is mythmaking, and the discussion of the Batman
Christopher Nolan:** One of the things I’ve enjoyed about working with
these characters is that they have the potential to be topical. And
the reason for that is they’re not real. It’s not real life. You’re
dealing with a heightened reality. You’re not dealing with Chicago or
New York; you’re dealing with Gotham City. And that gives you a very
interesting world to be able to play with in a very heightened way, in
a very operatic way.
And some practical reasoning from the Chicago Tribune:
Chicago, don't take this personally, but: "The Dark Knight Rises," the
first of Nolan's Batman movies not to be shot here, was wise not to
shoot here this time. For variety's sake, sure. But also, a better
reason: A quick shot of the Chicago skyline screams money and
prosperity. A quick shot of Pittsburgh, where a lot of the film was
shot, reveals working-class areas within yearning range of gleaming
skyscrapers. You'd have to find a pretty wide lens before you could
hold the poor of Chicago and the rich of Chicago within the same
frame. True, in "Rises" the Manhattan skyline is liberally mixed in;
all the better to play up the 1-percenters. (Still, two nods to
Chicago do sneak in, sorta: Batman swings by a historic neighborhood
named "Old Town," and those Gotham license plates are hilariously
reminiscent of Illinois license plates.)