Furnace is an item in The Legend of Zelda: Breath of the Wild. It is a furnace that powers Ancient technology when lit with a Blue Flame. There are two furnaces in Breath of the Wild used to power the Hateno Ancient Tech Lab and Akkala Ancient Tech Lab.
Furnaces appear during the quest "Locked Mementos" at the Hateno Ancient Tech Lab and the side quest "Robbie's Research" at the Akkala Ancient Tech Lab. They are used by the Sheikah Ancient technology researchers Purah and Robbie to power their respective labs. However unlike the Ancient Furnaces, the Furnaces' flames tend to burn out eventually, forcing Purah and Robbie to occasionally relight them. However as Robbie and Purah aged they had trouble lighting their furnaces, though Purah managed to reverse her age with an experimental rune and for a time she was about to relight it herself, but after she reverted to a six year old, she could no longer do it herself and she couldn't rely on her assistant Symin as he turned out to be unsuited to the task, forcing Purah to try and light it herself, only to be spotted by one of the Hateno Village children, forcing her to remain in her lab after the children began spying on her lab due to the rumors of a young Sheikah girl living at the lab. Purah was forced to rely on gas powered lights and was unable to power her lab's Guidance Stone and Travel Gate. Robbie however for a time was able to rely on his younger assistant Jerrin to relight his lab's furnace to power the modified Guidance Stone, Cherry. However after Jerrin married Robbie, she became jealous of the attention he paid to Cherry who was also named after Robbie's first love which only intensified Jerrin's jealousy to the point she even refused to relight the furnace to restore power to Cherry. However Robbie was for a time able to rely on his and Jerrin's son Granté to relight the furnace's though eventually Granté left home to see the world and Robbie was unable to relight the flame due to Jerrin's refusing to do so and his old age.
As a result, Purah and Robbie both request Link relight their furnaces for them to restore power to their lab's equipment. Link can acquire the Blue Flame from the Ancient Furnaces in Hateno Village and Tumlea Heights using a Torch. However he can not do this if it is raining forcing him to wait for the weather to clear by waiting or passing the time. Though the Ancient Furnaces are located far from each labs' furnaces, Purah and Robbie have apparently set up special Torch-like Lanterns that can be lit with blue flame. Once lit these lanterns remain lit and will not go out if it rains allowing Link to return to them to relight his torch with Blue Flame. Once he reaches the lab's furnace he simply has to light it with either a Torch, wooden weapon, or Arrow lit with blue flame to restore power to its respective lab. This will also cause the lab's Travel Gate to activate and be added to the Sheikah Slate as a Fast Travel point allowing Link to warp to the Tech Labs whenever he wants. This will also restore power to Purah's Guidance Stone allowing Link to repair his Sheikah Slate and Cherry allowing Link to purchase and forge Ancient Soldier Gear. The furnaces will remain lit for the rest of the game. Link is unable to lit the furnaces before unless he is taking on their respective quests, however he can light the lanterns with Blue Flame to make light the lab's furnaces later easier.