Skipping the Hero's Bow is by far one of the most difficult item to avoid. Even if you manage to skip the bow in Tower of the Gods, you will still obtain it once you obtain the Fire and Ice arrows and Light arrows. Therefore, you must skip the bow in Tower of the Gods, and all enhancements associated with it.
In Any%, there are two supposedly required uses for the bow : one to defeat Phantom Ganon in Ganon's Tower, and one to defeat Puppet Ganon. Both can be skipped, as seen on this page.
In categories that do more dungeons, the bow is also required in order to progress and defeat the boss of Tower of the Gods, and in order to enter Ice Ring and Fire Mountain for Iron Boots and Power Bracelets. These can also all be skipped using difficult tricks. Note that if Ice Arrows are skipped, then Medli is required in Earth Temple, meaning Hyrule 2 Skip cannot be performed.
The only requirement the bow has in Tower of the Gods is shooting the eye switch in the room right after obtaining the bow itself. Shooting the eye reveals movable platforms, which can then be used to carry the second statue across the gap into the central room.
This can be avoided with the use of chest storage. First, obtain chest storage in the first floor or from the tingle chest. Then, after you collect the second statue and carry it to the room where the bow is normally required, use chest storage to navigate around and through the pillars to carry it across the gap.
Normally to defeat Gohdan you must use the bow to shoot the hands and the eyes to be able to reveal his mouth which you then throw a bomb to deal damage. However, bombs also damage the eyes and hands, and can be used exclusively for the entire fight. Throwing the bombs at the hands and eyes is easier said then done though, as their movement is random and the bomb throw timing is precise.
It should also be noted that because you never obtained the bow, Gohdan will continuously snot out arrow pickups, as he thinks you currently have 0 arrows. This makes it extremely difficult to time the bombs and deal damage. This can be avoided by obtaining an arrow upgrade from either Western Fairy Island or Thorned Fairy Island. These upgrades do not give you the bow itself but set your arrow count to 60, so Gohdan will not snot arrows constantly.
In order to skip the Fire and Ice arrows, you must obtain the Power Bracelets and Iron Boots without their use. It should also be noted that this is only possible on the Japanese version. If you attempt to enter Fire Mountain or Ice Ring Isle without freezing or cooling them on any other version you will void out immediately upon entry.
To get into Fire Mountain without Ice arrows is fairly straight forward. Just start a hover from the edge of the island or from the platform and heal with either a heart drop from the Kargaroc or with the tingle tuner. If you timed healing correctly you will fall into the hole and be able to obtain the Power Bracelets.
Getting into Ice Ring Isle without Fire Arrows is one of the most challenge portions of the bow skip. There are two ways to do so.
The first method is to perform a tingle balloon superswim to get inside the island before it fully loads. Then wait for the island to load and jump into the loading zone
The second method requires Door Cancel, so you will be completely blind for the entirety of the skip. You will also need a lot of height, so you must start a zombie hover from Southern Fairy Island or another close island. Once close to Ice Ring Isle, heal yourself with the tingle tuner and leaf to the loading zone very precisely, as it's still very easy to freeze and void out.
A third method is to use Actor Unloading to unload the actor that causes you to freeze while walking on Ice Ring.
There are several ways to skip the light arrows. Only some of them are viable for low% though, because the bottle and the boomerang are also skippable.
Head over to the room on the right that leads to the Phantom Ganon battle and wait for the cutscene to finish. Head over to the left near the drop off by by the stone, and jumpslash on top of the stone.
Set a bomb down and then pull out the wind waker to perform a wind waker dive off of the bomb followed by dry storage. Store the text from the stone and run back over to the door. As you press A to open the door, the text from the stone will pop back up. Quickly mash through the text box to keep the door from closing. Now run over to the door on the opposite side of the room, watch the cutscene, and return to the preceding room. Run through the open doorway and leaf pump to the loading zone.
You can also skip the cutscene in the dark portal room by getting storage a second time on the stone tablet.
This method allows for skipping both the leaf and the bow on the japanese version, without the use of Actor Unloading, but it requires an item such as the boomerang to open the portal.
First, open the portal. Go to the room where the entrance to the maze lies and get storage off the stone tablet. Store the text and open the door. Close the text as soon as it appears to keep the door open. Get Storage again and store the stone tablets text again. Enter the portal room and make sure the door closes behind you. Close the textbox to get storage. Move KoRL sideways and then store his text. With his text, store the door. Then simply reload the room by going in and out of the room with the stone tablet. This will unload the portal room, but KoRL's position will be the same as when you left, so you can hookshot to him and sail to the loading zone, skipping the Light Arrows without the Deku Leaf.
One of the consequences of skipping Light arrows is you have no way of killing Puppet Ganon. Therefore, as a conclusion to the bow skip, Puppet ganon skip must also be completed. See here for how to skip him.