Is The Hunger Games a future or alternate world?

If it "used" to be called North America, then it's a safe bet that the reference to past-tense means it's a future version of our planet.

From my understanding (and this site, and this site), it's set in a post-apocalyptic North America in our own universe.

According to this interview with TIME magazine with Suzanne Collins ...

The Hunger Games is set far in the future.

and in this interview (discussing whether she would cameo in the films)

Collins is staying off-camera. "It's (set) in the future and to be in it would be alarming. I have no need to do a cameo."

I think we can safely assume she means the far future of our own world.

The Hunger Games is annoyingly bereft on any details about the world before "modern day" Panem.

We barely get any information about Before The Dark Days, (The first rebellion of the Districts against the Capitol, which lead to the Treaty of the Treason, and the Hunger Games being formed 74 years before the events of the books/movies). We don't even know about the rest of the world, let alone before Panem. All we have is the Mayor's speech, via Katniss' internal monologue.

He tells of the history of Panem, the country that rose up out of the ashes that was once called North America. He lists the disasters, the droughts, the storms, the fires, the encroaching seas that swallowed up so much of the land, the brutal war for what little sustenance remained. the result was Panem, a shining Capitol ringed by thirteen districts, which brought peace and prosperity to its citizens. Then came the Dark Days, the uprising of the districts against the Capitol. Twelve were defeated, the thirteenth obliterated. The Treaty of Treason gaves us the new laws to guarantee peace and, as our yearly reminder that the Dark Days must never be repeated, it gave us the Hunger Games."

Aside from that, all we know is that Rome/Latin existed, and that the Capitol is in the Rockies. Specifically named the Rockies.

[It isn’t until Mockingjay however that we learn the ironic meaning of “Panem.” I had thought the name was a variation of Pan-America, and maybe in one sense it is. As I had forgotten, District 12 was once Appalachia, and the Capitol was in the place “once called the Rockies.” [HG, 47] But we find in Mockingjay there is a more sinister association when Plutarch explains it to Katniss:]

It’s a saying from thousands of years ago, written in a language called Latin about a place called Rome, he explains. “Panem et Circenses translates into ‘Bread and Circuses.’ The writer was saying that in return for full bellies and entertainment, his people had given up their political responsibilities and therefore their power. [MJ, 223]

The Rockies weren't called that until late 1700s or early 1800s depending on your source, as western (mis)translations of the indian name for the mountains.

Neither of these are concrete evidence against a divergent timeline, but considering that there would be no point in considering it one without any evidence to that effect. Here, the absence of evidence is evidence to an absence thereof. Had Suzanne Collins intended it to not be our future, she would have made a point to mention it. In a interview with Scholastic, she specifically points out the timeline though:

Q: How long would it take for North America to deteriorate into the world depicted in the books?
A: You’d have to allow for the collapse of civilization as we know it, the emergence of Panem, a rebellion, and seventy-four years of the Hunger Games. We’re talking triple digits.

So Panem is minimum 200+ years into the future of North America. No correction about the question, no hemming and hawing. Frankly, as a story set a few centuries into the future, there is no need to change the back story of the setting, in somehow changing the past of the real world.

Furthermore, in that same interview, she compared Roman entertainment (Gladiators) to modern entertainment (Reality TV), and in the story, she uses the cliche of those who ignore history are doomed to repeat it, by having Coin trying to establish a Hunger Games for the Capitol's children. The literary theme would be ruined had something significantly changed the background of the story.