My understanding is that a lot of Templar assets were either conferred upon other orders such as the Knights Hospitaller, or were allowed to be used by Knights transitioning into a purely monastic life. This begs the question, was there ever a large stash of treasure which did not contribute to the aforementioned, and just disappeared following the order's demise?
While it is true that the Templars had a large amount of wealth, most of it was not liquid assets. Their most valuable assets were in the form of lands for farming, partly to finance their operations in the Holy Land and partly to support said operations with grains, cattle and mounts.
The other valuable asset they had was manpower. The Knights themselves were a minority, often coming from nobility like their peers in the secular world. What they didn't lack was their auxiliary, semi-ordained Sergeants; drawn from less prestigious background, they were the backbone of the Templars as soldiers and labourers. Combined with their established network in Middle East, the trained soldiers would make the perfect guardians of shipments and caravans between Occident and Orient. It would evolve, over the decades, to the safekeeping of valuables of noblemen throughout the Holy Land and Europe. It would in turn open the way to some sort of proto-banking where the Knights would loan capital (interest free of course, although they never refused a charitable donation ;)) against collateral. The same way banks today use the liquid assets of its clients to loan money.
It's important to note, however, that at the dissolution of the Order these assets were not hidden or stashed somewhere. In France, Philippe IV incorporated the assets into the Crown's estate. Outside of the kingdom, the Church took back most of it, to either keep it for themselves or give it to other orders (notably to the Hospitaliers, but also to the orders responsible for the Reconquista).
It is also important to know that most of what the Templars did and the wealth they amassed was reserved for two precise goal: protecting or reclaiming the Holy Sites and being the armed fist of the Pope (not the Church). They rarely if ever kept their wealth for themselves.
Source : Demurger, Alain. Les Templiers : Une chevalerie chrétienne au Moyen Âge, Paris, Éditions du Seuil, 2005, 664 p.
I hope it's alright by the almighty mods if my source is not in english! Demurger, as far as I know, is one of the topmost French expert on the Crusades.
It's true that most of the wealth was in land, and was therefore conferred to other orders. However, a couple years before the 1307 arrest, King Philip the Fair devalued the currency in order to pay debts and the populous rioted. He took refuge in the Paris Temple, where he first saw and longed to seize the gold in the Templar vaults. The vast wealth was stolen by the various kings after the arrests. There are a few other theories mentioned in a new narrative account of the Templars (fiction but probably based on the other works mentioned in this thread). It's a quick read: https://www.amazon.com/Knight-Temple-Brennan-Smith-ebook/dp/B01N0VYSI3/ref=la_B01N4JESDB_1_1?s=books&ie=UTF8&qid=1483735790&sr=1-1