LTV predicts that fiat currency, which requires zero labor to produce, should have zero value. Empirically, fiat currency does not have zero value. How do Marxists account for this. The to responses I have gotten so far are that labor does go into printing(though very little in comparison to the price) and that fiat currency is merely a token of money. In which case what is the money in this regard and how can any of this be empirically tested?
Marx's theory of product-value is often called a "labour theory of value" though he never used it himself to describe his "law of value". He also had a theory of exchange-value to try to explain market phenomena.
Marx did have a theory of the materialisation of labour in gold. But fiat or credit money has no intrinsic value (if it did then it would not be fiat), though it does have an exchange value, broadly speaking the value that can be acquired by spending it. Marx argued that the use of money obscured the real value ("congealed labour-time") involved in transactions and exchanges.
That is not really testable, but what may be testable is the extent that relative prices of different commodities reflect the labour involved in their production, and if so whether these reflections are of average or marginal labour, and what causes changes in relative prices over time. Whether you would want to is another question: it is all a bit 19th century.
There is nothing which exists in the human world, which does not require labor. Fiat money is by far no exception to that. It requires an state with many institutions to secure the money, so people can trust in the value of it. It requires bank houses and a lot of technology and real work to manage the financial processes. Marxian economics is very materialistic. The material background of most "pure" econonomic phenomens, is often ignored by the mainstream. For marxian economics fiat money has definitely value of "congealed labour-time" and not only the value of the printed paper. Today we live in the age of information and data. So we know a lot things similar to fiat money. How a society determines the value of something is in the power of the society, not in the thing itself. The average required labour time is an theoretical more philosophical tool, which still is not easily applicable to known, maybe any, economy.