Cinna dies because he showed rebellion towards the Capitol. He was designing dresses for Katniss that showed her off to be the Mockinjay, the symbol of the Rebels. For his rebellion, he was beaten in front of Katniss and then executed.
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doesn't say directly why but i think because katniss is considered and a rebel by the capital and they want to kill Cinna to get inside of her head and psych her out
Cinna is tortured by the Peacekeepers on pages 262 to 263.
you will be killed
Katniss, Peeta, Haymitch, Finnick, Betee, Johanna, Cinna, Gale, Thread, President Snow
Cinna is killed in Catcting Fire but his death is confirmed in Mockingjay. As Katniss is about to enter the arena once again for the third annual quarter quell, a few peacekeepers come in and beat Cinna to death. Katniss can't do anything but scream because she is in the glass cylinder. Cinna was my favorite character after Rue got killed so of course I was sad.
The book after Catching Fire is Mockingjay.