Fireweaving is using an off GCD ability(swiftcast, surecast, manawall, manaward, raging strikes, apopopopostasis etc) right after a Fire I to see if you got a Firestarter proc and hit it before casting another Fire I. This isn't particularily useful, other than on the last Fire 1 in your rotation. If it procs at that point, you're already casting Blizzard III, so you'll have to use it at the end of the mana regen rotation, with Transpose, meaning your Fire III will only get 1stack of Astral Fire damage bonus. If however you fireweave on that last Fire I, you can then hit the Fire III at 3stacks, then Blizzard III.
The gain is minimal, and if it doesn't proc, you just waste time, and a cooldown(which might or might not be relevant, depends on content). If it never procs over the course of a fight, you will do less damage than if you hadn't bothered at all. Same if it doesn't proc enough. It needs to proc several times and you need to catch the procs right away for it to be worth doing, and you're still potentially losing utility from dumping random abilities at the end of a rotation.
Out of the BLM tricks, it's probably the least useful.