What would happen if a game went on too long? Tom Doyle has answered his own question - "The gamemakers could kill off the victors and drive them together", as we see done in the first book:
The Hunger Games (book 1), Part II "The Games", Ch. 13:
It’s not hard to follow the Gamemakers’ motivation. There is the
Career pack and then there are the rest of us, probably spread far and
thin across the arena. This fire is designed to flush us out, to drive
us together. It may not be the most original device I’ve seen, but
it’s very, very effective.
The second part of the OP's question, "What would happen if this failed to work and they still didn't kill or attack each other"? If, say, the tributes from the outer districts were to team up and kill the highly aggressive inner district tributes, and then just sit down and refuse to fight? The Capitol would, first, execute them, and second, severely punish their families (at least), maybe even everyone in the districts they came from. The tributes' families, friends and people of the districts are the hostages that ensure the tributes put on a good show. While this doesn't seem to be expressly stated, there's enough circumstantial evidence to indicate the Capitol's method of operation.
The Hunger Games (book 1), Part I "The Tributes, Ch. 8:
At first, I expect guards to come for me. But as time passes, it seems
less likely. I calm down. They still need a girl tribute from District
12, don’t they? If the Gamemakers want to punish me, they can do it
publicly. Wait until I’m in the arena and sic starving wild animals on
me. You can bet they’ll make sure I don’t have a bow and arrow to
The Hunger Games (book 1), Part I "The Tributes", Ch. 8:
“Do you think they’ll arrest me?” I ask.
“Doubt it. Be a pain to replace you at this stage,” says Haymitch.
“What about my family?” I say. “Will they punish them?”
“Don’t think so. Wouldn’t make much sense. See, they’d have to reveal
what happened in the Training Center for it to have any worthwhile
effect on the population. People would need to know what you did. But
they can’t since it’s secret, so it’d be a waste of effort,” says
Haymitch. “More likely they’ll make your life hell in the arena.”
The Hunger Games (book 1), Part III "The Victor", Ch. 21:
I nod because I do understand. About owing. About hating it. I
understand that if Thresh wins, he’ll have to go back and face a
district that has already broken all the rules to thank me, and he is
breaking the rules to thank me, too.
Judging from the general tone of the other questions that people ask, I think there's a very basic misunderstanding about the rules of the games. The "rules" are whatever the Capitol says they are. If the Capitol wants to "break" the "rules", there's no higher authority that can be appealed to.
The Capitol could simply march into the districts, grab two kids at random and shoot them on the spot. They came up with the games as a way to, first, legitimize what is in fact just an execution, and second, display their technological superiority and not just their brute strength.