It appears from the movie (never read the book) that the Hunger Games are shown "live". That there is no delay in the broadcast. It appears to be shown live as the audience is shown watching scenes that the Gamemakers and the Capitol would never want them to see. For example, after Rue is killed, Katniss throws up the sign for "respect" which causes the uprising of district 11. Why show that scene at all in a taped or delayed broadcast?
In the book at least, it appears the Capitol can and does cut out scenes they don't want shown:
The Hunger Games (Book 1), Part II "The Games", Ch. 18:
Slowly, one stem at a time, I decorate her body in the flowers. Covering the ugly wound. Wreathing her face. Weaving her hair with bright colors.
They’ll have to show it. Or, even if they choose to turn the cameras elsewhere at this moment, they’ll have to bring them back when they collect the bodies and everyone will see her then and know I did it. I step back and take a last look at Rue. She could really be asleep in that meadow after all.
“Bye, Rue,” I whisper. I press the three middle fingers of my left hand against my lips and hold them out in her direction. Then I walk away without looking back.
The Hunger Games (Book 1), Part III "The Victor", Ch. 27:
They play her death in full, the spearing, my failed rescue attempt, my arrow through the boy from District 1’s throat, Rue drawing her last breath in my arms. And the song. I get to sing every note of the song. Something inside me shuts down and I’m too numb to feel anything. It’s like watching complete strangers in another Hunger Games. But I do notice they omit the part where I covered her in flowers.
Right. Because even that smacks of rebellion.
The capitol didn't show the act of Katniss covering Rue's body with flowers or her salute. But the floral burial must have been shown when the bodies were picked up. It's not covered in the book, but while the actions of the tributes are monitored and edited for possible seditious acts, the recovery of bodies might be watched less carefully.
Per the Hunger Games wikia, gambling is one of the primary reasons for the popularity of the show. This includes "in-play betting" such as on the outcome of individual encounters.
Introducing a broadcast delay would allow unfair opportunities for those with insider knowledge.
The Hunger Games is a major source of gambling and produce intense betting, both in the Capitol and the districts. Katniss mentions people from District 12 betting on the two tributes reaped, and that "odds are given on their ages, whether they're Seam or merchant, and if they will break down and weep." In the Capitol, betting takes place throughout the Games, starting before training and increasing in intensity until a victor is determined.
The "Hunger Games" are shown live because, like @Richard said, there is gambling involved (not going to go further this road).
But there is also the sponsors. If it wouldn't be shown live, then how would the sonsors have any impact in the games?
For example, in the 1st hunger games Katniss is sent a cream to help heal her fire wounds, and in the second she is sent a sort of tap take enables her to get water. If they were not shown live, then these gifts, that from what the books say cost a lot of money, wouldn't come in time.
The reason the poor districts were rioting was because they were oppressed and hungry. Cutting scenes from a show will not make them less oppressed or less hungry.
Censors assume they can control the thoughts of the audience. There is no sure way to control the way your audience thinks.
For me, all of Katniss' choices are a slap in the face of the capitol and its policies
choosing to ally herself with a weak contestant (Rue)
choosing to help the weak
choosing to protect and take care of the young
choosing to share her food even tho she was also hungry
The only probable way to censor her is to not show her at all, and even if they don't show her I doubt it will make the poor districts less hungry or less oppressed.
from book 1
Taking the kids from our districts forcing them to kill one another while we watch- this is the Capitol's way of reminding us how totally we are at their mercy. How little chance we would stand of surviving another rebellion.
Whatever words they use, the real message is clear. "Look how we take your children and sacrifice them and there's nothing you can do. If you lift a finger, we will destroy every last one of you. Just as we did in District Thirteen."