Any questions that might be asked about literature or an issue after having read The Hunger Games will be subject to and dependent on the reader. Beatty from Fahrenheit 451 supported the burning of books because in his opinion books only offered more questions and brought further confusion to people. The critical questions that might be asked from any piece of literature are incredibly dependent on the reader and their personal worldview. With that said, I'll attempt to highlight a few critical questions I believe The Hunger Games touched upon.
Issue #1: Society's general assumption and fixation on the idea that our future society is going to hell in a hand basket. The world that Katniss Everdeen lives in is not a bright future. Many of the 12 districts are struggling for survival and most of what they produce is ripped from their borders by the greed of the Capitol. Gone is the technology and infrastructure that we currently have. Add to that the fact that an annual arena of death featuring children is considered normal and necessary. That's one messed up society. But it is an issue and concept that is repeated over and over again in Hollywood through movies like "Mad Max," "The Postman," and "Waterworld."
Issue #2: A female character uses "love" to most benefit herself. Gale even calls Katniss out on this one. If it benefits her to string Peta along, great. If she needs the help from Gale, then string him along for a bit. This is nothing new in modern teen literature featuring a female lead and two doting males. Consider, for example, Bella Swan from the Twilight series. Depending on the book or part of the book, Bella just can't decide who she needs more—Edward or Jacob. And that's why too many websites are devoted to Team Edward or Team Jacob.
Issue #3: Are females finally emerging as strong hero character types? Previously, in books, TV, video games, etc., the hero was generally three things—young, white, and male. Luke Skywalker, Peter Parker, Harry Potter, Captain America—the list goes on and on. Very few highly successful franchises focus on a female lead. Tomb Raider's Lara Croft would probably come closest (but she is also held subject to the standard beauty myth as well). Readers could make the argument that Bella Swan and Katniss Everdeen are breaking the male hero mold. I would argue differently, though, because they are only breaking 1 of the 3 mold stereotypes. Instead of being young white males, they are young white females.
Issue #4. Portrayals of violence. Is society becoming more fixated on violence? Less? Same? The Hunger Games is a violent book. There's no doubt about it. So are young people being exposed to more of it than they were a few generations ago? I could argue either way. The Roman Colosseum was quite violent and not that much different in concept than The Hunger Games. . . except that it was real.
I hope it helps!