After reaching the Hateno Ancient Tech Lab and speaking with the director inside, you will be asked to obtain an ancient blue flame to light the furnace. The blue flame is just down the hill, but I don’t know is how to bring it back up. The sticks keep going out of breaking before I can get close. How do you get the blue flame back to the ancient tech lab?
Well, the first thing you need is to get plenty of tree branches or torches as it will take a few of them to get the flame all the way up to the top of the hill. If you are short on branches see this post on how to get more tree branches. Take note of all of the lanterns around the town and in the walk up to the ancient tech lab. This is key to solving the puzzle.
Light a tree branch on the blue flame and begin following the path up to the tech lab. Make sure you light every lantern on the way as you will need to go back to the last lantern once your stick breaks. Relight a new stick on the most recent lantern if this happens. Eventually, you will have made your way back up to the top of the hill and be at the tech lab. The furnace can be found to the left of the front door.