How do Marxists account for fiat money? 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.