How do I solve the fire maze in the demon side?

The easiest way to get the chest, from the demon side, is from the beginning of the level, go to the side opposite of the ink well. Oswald will show an icon to give you a boost up to a ledge just beyond jumping range. Collect your goodies there, and then carefully jump out onto the boards holding the curtains of fire up. If you are careful, you can walk across the top of the entire thing and hop down right next to the chest.

I also seen somebody mention you can thin the crystals near the inkwell instead to turn the ink yellow and then walk through the fire instead.

You also could go up to Angel side, throw Oswald up near the bridge and float down with him to demon side and land in the spot with the chest. If you go this route, and then exit through the demon side D.E.C you also complete the Fall From Grace challenge.

Use indelible ink to walk to the chest in the middle of the maze. From there, keep turning the same handle until all the fire is shut off, and you can explore.

Specifically, from the Angel side, drop anvils on as many monsters as you can. Using a clock gives you more time.
Now, look at the maze. Float down to the left side of the fire, where there is a well. The well has liquid that makes you invisible when things around it are painted, and makes you invincible when you thin it.
BEWARE, there is a monster inside the maze, but you can drop an anvil on him. Or use a clock and TV. Even if you die, after you get to the middle with the chest, you respawn there.

Explore near where you floated down, and thin the yellow crystals on the ledge, to have Oswald throw you up there. Explore below.
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When you cross over to the Devil face, there is a monster there. Use the clock, TV, and anvil.
Get up on the platform, and keep turning the handle to raise a little platform that allows you into the mouth of the statue.
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When you arrive on the other side, you can find and return using the projector screen, so you get all the films for the movie theater and quest/challenge.
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Most films are easiest to get when both ears are on the left side. If you put on a Steamboat Willie costume, you see a pin in the 2D cartoons.

If you cerebrally want to see if it is possible, I did the first time I landed down there.
Notice that only a few squares have a handle in them.
Near the ledge with the amber crystals, look at the maze.
There is one handle you can turn twice, to make a path forward inside to the next square.
From that square, turn the handle so you can go right.
From that square, go forward.
Then left, toward the chest.
Turning the handle at the chest shuts off all the fire.

At one place I heard a noise when I landed on an insignia in the center of the square.

BEWARE of the monster near the chest.
Drop clocks, TVs, and anvils.
Also, if you did not destroy the monster on the pedestal above, get him.

Go to the Angel Side Have Oswald use his helicopter Ears and jump on his legs from their you can land on top of the those brown pole things on the devil side and jump down when you get to the chest, the lever next to it will get rid of the fire walls.