I think the original book and the movie would certainly be appropriate for the average 16 year old. The violence is not meant to be glamorous, and unlike many other "action-adventure" type stories, even justifiable violence (self-defense, defense of others) is depicted as having lasting serious psychological effects not just on those who suffer violent acts, but those who commit the acts themselves. The heroine's actions wind up resulting in the deaths of many people, and she is affected by them, permanently and seriously, even though you could argue that her actions are justifiable for the most part.
The trilogy, taken as a whole, raises some tough questions on when violence is justified, and the last book in particular addresses weighty topics such as justifications for war, armed rebellion, assassinating government officials, etc. I personally think the author bit off a little more than she could chew in the third book, where she seems to be "preaching to the choir" of readers who already know that "war is hell" -- amazon.com has many rave reviews for the last book, "Mockingjay", from older readers who have experienced war or know people who have, but not as many from the actual target demographic for the series, young teenagers.
That being said, there are some readers and viewers of the Hunger Games who for some reason do not "get" this message and do indeed find it "cool" that a teenager could be forced to become a gladiator and kill people for entertainment.
There are also caveats regarding the romance in the series, as the heroine winds up in a love triangle between a childhood friend and a young man she met at the Games. There is no obvious intimate activity beyond passionate kissing, but there are subtle references to sexual feelings. The heroine at one point sleeps in the same bed as one of her suitors, which she herself doesn't see as a sexual act, but such behavior in real life obviously could lead to sex or at least the temptation for it. Also, the heroine at various times finds herself in situations where she believes that one of her love interests is trying to kill her, and at times she is seriously tempted to kill him. While these situations make sense in terms of the story, the whole idea that it is somehow romantic to love someone and still have urges to kill him or her, did strike me as somewhat reminscent of Twilight. Also,
In the third book, the heroine is criticized by another character for not "standing by her man" even though he's tried to kill her. Again, this makes sense in the context of the story because the only reason the guy tries to kill her is because he's brainwashed to do so by the villians in the story. However, in real life, this would certainly NOT be good advice to give a woman who is subject to violent behavior from an intimate partner.