Did Furman end up dying from the death penalty?
No, William Henry Furman, who was the appellant in this Supreme Court case, did not end up being executed. He is still alive, though he is in prison again.
In 1968, Furman killed a man while in the process of robbing his home. Furman was sentenced to death. In 1972, the Supreme Court of the United States struck down the law that allowed Furman to be sentenced to death. It held that Georgia’s death penalty law was essentially arbitrary and unfair. Georgia and other states had to rewrite their death penalty statutes to make them fairer.
As for Furman himself, he was not sentenced to death again. He was released from prison on parole in 1984. He is currently in prison again having been arrested while burglarizing a house in 2004. He is serving a 20 year sentence. You can see basic details about his current status in this link.
The first link below for the Georgia Department of Corrections is supposed to take you to a page about Furman, but it redirects to their home page instead, so I guess you actually have to do the search if you're interested. You click the "Find An Offender" button in the top right and enter his name.