In Mockingjay, President Snow sends out a pack of mutts threw the sewers to kill Katniss. The mutts are a half-human/lizard hybrid that smell like roses and blood (that's how President Snow smells). These mutts wont stop until they kill Katniss, but they'll kill anything that gets in the way. Well, Finnick and some soldiers from District 13 got in their way, and they were killed. Katniss, Peeta, Gale, Cressida and Pollux mannaged to escape, but Finnick died before he could see the birth of his son.
"Unable to accept it, I shine the light of Cressida's gun down the shaft. Far below, I can just make out Finnick, struggling to hang on as three mutts tear at him. As one yanks back his head to take the death bite, something bizarre happens. It's as if I'm Finnick, watching images of my life flash by. The mast of a boat, a silver parachute, Mag's laugh, a pink sky, Beetee's trident, Annie in her wedding dress, waves breaking over the rocks. Then its over. I slide the Holo from my belt and choke out ' nightlock, nightlock, nightlock.' Release it. Hunch against the wall with the others as the explosion rocks the platform and bits of mutt and human flesh shoot out of the pipe and shower us."
Katniss, Finnick, Boggs, and some other people set out to the Capitol to kill President Snow. These lizardy mutts chase them and eat Finnick and Boggs. Poor Annie. She was married to Finnick in District 13.
As the mutts (sent from President Snow) are chasing Peeta, Gale, Finnick, Katniss, and everyone else who is alive....the mutts get to Finnick and he starts to scream. Katniss attempts to save him but there were too many mutts so she used the holo and shouted "NIGHTLOCK NIGHTLOCK NIGHTLOCK" and threw the holo down the ladder and killed Finnick and some of the mutts
Finnick, along with Katniss, Peeta, Leeg 1, Leeg 2, Jackson, Cressida, Boggs, and a few others(Not including Johanna) went on a mission which Katniss and Finnick jokingly called the 76th Hunger Games. Finnick was killed when he stayed to defend against the mutts at the Meat Grinder. I don't remember if he was decapitated or something else. After his death, Annie went into a deeper depression.