From what I understand. LTV predicts that fiat currency, which requires zero labor to produce, should have zero value. Empirically, fiat currency does not have zero value. Would this prove that the labor theory of value is wrong?
Value "in use" is the usefulness of this commodity, its utility. A classical paradox often comes up when considering this type of value. In the words of Adam Smith:
The word value, it is to be observed, has two different meanings, and sometimes expresses the utility of some particular object, and sometimes the power of purchasing other goods which the possession of that object conveys. The one may be called "value in use"; the other, "value in exchange." The things which have the greatest value in use have frequently little or no value in exchange; and, on the contrary, those which have the greatest value in exchange have frequently little or no value in use. Nothing is more useful than water: but it will purchase scarce anything; scarce anything can be had in exchange for it. A diamond, on the contrary, has scarce any value in use; but a very great quantity of other goods may frequently be had in exchange for it (Wealth of Nations Book 1, chapter IV).
Value "in exchange" is the relative proportion with which this commodity exchanges for another commodity (in other words, its price in the case of money). It is relative to labor as explained by Adam Smith:
The value of any commodity, [...] to the person who possesses it, and who means not to use or consume it himself, but to exchange it for other commodities, is equal to the quantity of labour which it enables him to purchase or command. Labour, therefore, is the real measure of the exchangeable value of all commodities (Wealth of Nations Book 1, chapter V).
Value (without qualification) is the labor embodied in a commodity under a given structure of production. Marx defined the value of the commodity by the third definition. In his terms, value is the 'socially necessary abstract labor' embodied in a commodity. To David Ricardo and other classical economists, this definition serves as a measure of "real cost", "absolute value", or a "measure of value" invariable under changes in distribution and technology.
Ricardo, other classical economists and Marx began their expositions with the assumption that value in exchange was equal to or proportional to this labor value. They thought this was a good assumption from which to explore the dynamics of development in capitalist societies. Other supporters of the labor theory of value used the word "value" in the second sense to represent "exchange value"
So No, fiat currency has exchange value which is equal to the amount of labor you can purchase with it.
Mandel claims Marx treats paper money as a sign of the underlying money commodity:
Paper money, banks notes, are a money sign representing a given quantity of the money-commodity. Starting from the above-mentioned example, a banknote of £1 represents 1/10 ounce of gold. This is an objective ‘fact of life’, which no government or monetary authority can arbitrarily alter. It follows that any emission of paper money in excess of that given proportion will automatically lead to an increase in the general price level, always other things remaining equal. If £1 suddenly represents only 1/20 ounce of gold, because paper money circulation has doubled without a significant increase in the total labour time spent in the economy, then the price level will tend to double too. The value of 1/10 ounce of gold remains equal to the value of 10 quarters of wheat. But as 1/10 ounce of gold is now represented by £2 in paper banknotes instead of being represented by £1, the price of wheat will move from £1 to £2 for 10 quarters (from two shillings to four shillings a quarter before the introduction of the decimal system).
However, price and value aren’t the same:
The price-form, however, is not only compatible with the possibility of a quantitative incongruity between magnitude of value and price, i.e., between the former and its expression in money, but it may also conceal a qualitative inconsistency, so much so, that, although money is nothing but the value-form of commodities, price ceases altogether to express value. Objects that in themselves are no commodities, such as conscience, honour, &c., are capable of being offered for sale by their holders, and of thus acquiring, through their price, the form of commodities. Hence an object may have a price without having value. The price in that case is imaginary, like certain quantities in mathematics. On the other hand, the imaginary price-form may sometimes conceal either a direct or indirect real value-relation; for instance, the price of uncultivated land, which is without value, because no human labour has been incorporated in it. (Capital 1 3:1)
Fiat currency, the object itself such as a dollar bill, indeed takes labor and capital to be produced. People work with printers to make those baby greenies.
I think your question has a few problems. What is the definition of LTV you're working with, for example? What exactly is value? These are very broad and abstract questions most economists nowadays just don't care about anymore. I guess we're more interested in solving object problems such as poverty and hunger more than in discussing the sex of angels, fortunately.
Wikipedia defines LTV as:
The labor theory of value (LTV) is a theory of value that argues that the economic value of a good or service is determined by the total amount of "socially necessary labor" required to produce it, rather than by the use or pleasure its owner gets from it (demand) and its scarcity value (supply). It does not say that the value of a commodity is determined by the actual amount of labor contained in it, but the average amount needed to produce it. This is called "socially necessary labor".
Due to the fact that all currencies roughly (average) cost the same to make yet have wildly different values across countries must mean that LTV's theory that "average amount" of labour needed to produce notes cannot correlate with its value whatsoever, and therefore the value of currency must entirely be subject to supply and demand of currency.
As the same would not be true with global commodities like gold, this is evidence to suggest that fiat currencies do indeed disprove LTV.
However, if one views the cost to produce money/notes (i.e. the cost to central banks from mints) rather than their fiat value, then LTV is not disproven as the 'socially necessary labour' is relatively equal globally and has no connection to currency values.
Note: I may have misunderstood your question but I had a go