This is sort of a small or mini guide, but I felt some of this stuff deserves to be mentioned. In this day and age where Sky Tower is just nuke the boss and repeat for the most part, there's still some areas where Sky Tower clear time can be (sometimes greatly) impacted that I often don't see happening, and so this guide is meant to serve as a sort of extra tips and tricks sort of thing for players who already know the basics of Sky Tower.
Note that I will not go into covering how each floor operates! That is not this guide's intent. Although it's outdated, this guide here does that.
With that being said, let's move on with some unique things you can do to speed up some floors, even if you're not bringing a ton of damage! Nobody is useless.
Floors 9, 27, 29, and possibly others: Remember those eidolons with the useless star buffs of a 5% and 10% chance of zeal? On floors such as these, where the damage seems more static and unaffected by your damage, crit damage, elemental damage, etc., these eidolons can be useful (which would mean all zeal sources would be).
Floor 9: On this floor, there's something interesting you can do. The large fire mobs around the edge of the room are not bosses, and therefore not immune to the knockback effect. Since there's such a large distance between the water pool and them, what this means is if you get a handful of people dedicated to knocking them closer to the center of the room, this can speed this floor up. To get the most out of this, it helps to have at least, say, three or four players doing this. Don't have that many Wizards and/or Grenadiers in your raid? Inefficient! Drop a few Lancers or Shurikens or something; DPS for Sky Tower is a dime a dozen. ^^;
While anyone can technically simply do this with a weapon that has this effect, it's really only worth doing with two classes in particular, and those are the Wizard and/or Grenadier classes. The reason is, these two classes have a mastery that can supplement making one of your skills knock back the target, whereas the weapon effect is limited to a 20% chance. I do recommend getting a weapon with this effect on top of this (and in the case of the Wizard, inlaying the basic secret stone that extends the range of Fireball), but it's not worth doing really unless you are one of these two classes using the mastery. Grenadiers and Wizards aren't useless after all!
Floor 18: Many people don't seem to know that the orbs that spawn that transform Hel always spawn in the same spot! They drift around a bit once they are there, but they always spawn where it last was.
What is the implication of this? It means if once you get Hel to an orb and transform her, once her elemental vision has been defeated and her normal form returns, everyone should stand under her and not move as much as possible. Use combo skills through her ability that makes you dance, or simply let it hit you and heal after (or if you have Bards, have them heal). This way, as soon as the orbs return, they will return right where she is, and she will transform again. In effect, that makes clearing this floor as fast as you possibly can (DPS aside).
Instead, players often run about randomly, and Hel is constantly jumping around the room with her random aggro, sometimes making it frustrating for that one player who is trying to lure her. This is a case where five high DPS players following this method will clear faster than those same five players and 35 others who are not. Try instructing everyone to wait the floor out and try it, and you will actually clear faster.
Floor 19: This one is less a secret maybe, but still not commonly done (or maybe known), and it's the difference between completing this floor in two minutes, or potentially up to ten. Like floor 18, a lot of time is lost here with most raids when it doesn't need to be so, but unlike floor 18 this needs everyone's participation. As soon as the Golden orb spawns, every... single... player... (no exceptions) should grab this and rush the free glimmer you get at the start. If you have potions, set them to be used to help with your maximum HP going up and the attacks you start receiving once you pick a glimmer up. If you are fast, you can get it a second time. After that, have half or more of your players change to the Red buff to farm the field.
If your group is doing this correctly, you will notice you may be stuck around the glimmer drop off point waiting for them to merge to drop your own off more often than not. Have Bards stationed near the Glimmer return point to keep players healed. It's fine to throw a life or a few away here if needed, if it means speeding the floor up, as risk of losing all lives through floors 1 to 20 is low.
Floor 22: The first of the two bosses that spawns can be knocked back towards the wall (using Wizards and/or Grenadiers) or simply have everyone stand on the boss. Minor mention but a small speed up.
Floor 27: Like with floor 9, after you have cleared the mobs and the flying demons show up with the shields, you can attempt to displace (using knockback) all of these towards where the first one is that Ashe runs towards. Just be careful, as when you are not transformed into a robot, you take damage every so often, so make sure you can keep sustained somehow.
Floor 30: Killing Asche before Shirley can (sometimes?) end the floor and spawn the chest, regardless of if Shirley was defeated.
Additionally, a scarecrow boss spawns when this chest does, and this is an optional boss. You may ignore it, defeat the chest, and proceed to the next floor without defeating it. It does have drops but does not spawn another chest.
Floor 31: Not so much a tip or trick to speed it up, but it helps to be organized here. Have enough people killing and luring the mobs together, ideally away from the center. Also, those devoted to gathering and not attacking, please do remove your mainhand weapon, so that when you get the debuff from a soul that makes you jump and do AOE damage, it doesn't do much damage to yourself and surrounding raid members.
Floor 32: Make sure you let the mobs get close enough to her shield before killing them. This is another floor that can last less than a minute or nearly five, the latter when there's too much DPS nuking everything blindly. Guardians with their pull ability are nice here!
Floor 36: By using a transform item...
...you can get out of the forced mob transforms you are put into. This is useful during the second part of the floor, where you fight the boss.
There is also some trick with the Shuriken (perhaps also Duelist with the roll?) to get over the wall during the first part to finish that part faster, but I am not very familiar with the actual details of doing so, so if anyone knows, please advise!
Floor 38: This is another floor like 18, in that it can often be faster by excluding those who are unsure what to do. It might help to set up a party or two, each one having a Sorcerer (main or sub), and four others devoted to running, ideally with high move speed. Have all four stand on the one side of the room, get hit (ideally by the outer circle AOE skill), have the Sorcerer immediately cleanse them of the slow, and run to the other side and give her the skill before it is lost and summons mobs. It's not a super fast floor regardless, but this can prevent it from being a fifteen or twenty minute floor as well.
Floor 44: Most may be aware that there are four buffs that each cube has, and killing one cube removes one of those four buffs, generally making it easier to defeat subsequent cubes (especially after the ice cube is gone). Most may also be aware that the cubes can also not be debuffed... not normally, anyway. However, if the one with aggro of the cube "resets" it (that is, leaves the radius of it, which, don't worry, won't actually fully reset it's HP to full as it would normally; it'll only reset it's aggro state), then it can be debuffed. Things like damage taken, defense reduction, and all the other sorts of debuffs players may have can go a loooong way here. Please, please become aware of this. It greatly helps versus watching the same person retain aggro and not reset it (because it can't be debuffed) as, say, the ice cube barely takes damage. Once the ice cube is gone, it sort of matters a bit less (especially in higher DPS raids), but also be aware that the cubes will start respawning if you don't get rid of them fast enough.
I believe that's about it! I am sure there's other things, and I'm not super familiar with floors 36 and above in particular, so if anyone has their own to add, feel free to comment.