To enter Shadow Temple without Din's Fire, you will need to get on the high green ledge by doing Shadow early. Follow the ledge to directly above the stairs leading down to the area with the torches. If you look down you'll see the torches nor the door is loaded. Get to the very edge of the green ledge then sidehop down in the direction of the stairs. You should land at the bottom without loading the torches or the door, so just walk in the direction of the shadow temple and you'll enter it.
The room with the boss key has 2 spike walls closing in on you that you normally need to burn to access the Boss Key. To get through, as soon as you enter the room, run diagonally up right and run into the spike wall. Swinging your sword once you touch the wall also sometimes helps. If you're fast enough, the redead guarding the chest should freeze you while the spike walls continue to move. When you're unfrozen, you should be on the other side of the spike wall and will be able to grab the boss key. To get out, you'll either need to wait until the walls come together which will "crush" you and reset the room, or use Farore's Wind.
Unless you do the Trials Skip, you will always need at least one of the fire items. Fire arrows are generally the preferred item.
In the Shadow trial room of Ganon's Castle, there's a torch you need to light to activate white platforms to cross the room. If you have longshot, the easiest way past this room is longshot to the first torch to the right, then from there longshot to the likelike. Once you activate the Golden Gauntlets chest, longshot the chest to grab the treasure, then you can just finish the room off like normally.
If you don't have longshot, its a bit more complicated. You'll either need to hover boost with the megaton hammer or superslide to the first torch. If you have dins fire, you can light the torch from here then run off the edge with hover boots to grab a platform and finish the room. If not, you'll have to do a very tricky superslide to the next platform. Hovering is also an option, but obviously not very practical.
At the last part, there is a web you must burn to shine light down and use the mirror shield to shine light on a sun to open the door. This can be done by getting near one of the torches next to the crystal switch that opened the last door and shooting a regular arrow through the torch at the web. Its a bit of a tricky shot but possible.
This room is the sole reason for the note above that says at least one fire item is required. There is no way to light the torches with another fire source and no way to move on to the next room without lighting the torches. If you have Din's Fire, you can light the 4 bottom torches with it then shoot an arrow through one of them at the 5th highest torch. With fire arrows, obviously just shoot all the torches.