He was almost certainly killed.
Here's an excerpt from the end of Mockingjay. Katniss is being prepared for the execution scene. (Important note: in the novel Mockingjay, Effie Trinket appears only in this one scene; her role was considerably beefed up for the films. Katniss's prep team have been in District 13 throughout, but treated badly and even imprisoned at one point for taking more bread than their daily quota.)
In the bedroom, I find another surprise. Sitting upright in a chair. Polished from her metallic gold wig to her patent leather high heels, gripping a clipboard. Remarkably unchanged except for the vacant look in her eyes.
"Effie," I say.
"Hello, Katniss." She stands and kisses me on the cheek as if nothing has occurred since our last meeting, the night before the Quarter Quell. "Well, it looks like we've got another big, big, big day ahead of us. So why don't you start your prep and I'll just pop over and check on the arrangements."
"Okay," I say to her back.
"They say Plutarch and Haymitch had a hard time keeping her alive," comments Venia under her breath.
"She was imprisoned after your escape, so that helps."
It's quite a stretch. Effie Trinket, rebel. But I don't want Coin killing her, so I make a mental note to present her that way if asked. "I guess it's good Plutarch kidnapped you three after all."
"We're the only prep team still alive. And all the stylists from the Quarter Quell are dead," says Venia. She doesn't say who specifically killed them. I'm beginning to wonder if it matters.
So what do we know from these few paragraphs?
- The victorious rebels would have executed Effie Trinket if her case hadn't been hotly pleaded by people high up in the planning of the rebellion who knew her personally. In order to do so, they had to present her as a rebellious spirit (which, in the book, she wasn't) and lean heavily on the fact that the Capitol had imprisoned her.
- Not just Cinna but all the stylists from the last Hunger Games were dead - executed either by the Capitol or by the victorious rebels. It seems that, like that of a Hunger Games victor, the role of a stylist was a very dangerous one during the war: being suspected by each side of possible allegiance with the other, they get caught in the middle and end up dead either way.
But Effie Trinket and the Hunger Games stylists, although recognisable to a lot of the population, took no part in either side's propaganda efforts as far as we know. They were hardly high-profile targets for either the rebels or the Capitol. About the only thing they were guilty of was association with the Hunger Games and in particular the Quarter Quell. From the Capitol's point of view, they were associated with the Quarter Quell victors who helped destroy their Games; from the rebels', involvement with the Hunger Games system might be enough to consider them criminals.
Now let's look at Caesar Flickerman. He was very heavily involved with the Hunger Games - he was 'the face of the Games', so to speak. If Effie was a candidate for execution, Caesar must have been even more so. Not only that, but he was involved in most if not all of the Capitol's propaganda clips involving Peeta during the war. He was the face of the Capitol propos. If he was still alive by the time the rebels won, he would definitely have been on their list of people to execute.
And that's assuming the Capitol hadn't already killed him. The last time we see him in the books is in his last interview with Peeta, with its - from the Capitol's point of view - disastrous ending as Peeta breaks free of their mind control and warns District 13 about their impending attack. If Caesar got the blame for this failure, I'm sure Snow wouldn't have hesitated to have him killed. Unlike in the films, he doesn't appear to report Katniss's 'death':
Now we cut to a live feed. A reporter stands on the roof with the Peacekeepers. Behind her, the apartment block burns. Firefighters try to control the blaze with water hoses. We are pronounced dead.
We watch as they play the footage over and over. Revel in their victory, especially over me. Break away to do a montage of the Mockingjay's rise to rebel power - I think they've had this part prepared for a while, because it seems pretty polished - and then go live so a couple of reporters can discuss my well-deserved violent end. Later, they promise, Snow will make an official statement. The screen fades back to a glow.
All that's mentioned is anonymous "reporters". Given her personal experience of Caesar Flickerman, I think Katniss would have thought of him as himself rather than just a reporter if he'd appeared on the screen, so we can assume he doesn't appear in the reporting of what seems to be a victory for the Capitol. This is more evidence that he's out of action even before the end of the war.