I have several dollars on top of my dresser, but most of my money (in pretty much any sense in which economists regularly use the word) exists as electronically-recorded liabilities of financial institutions. For most of my bills, it is more convenient to pay them out of such intangible money than the tangible money. Supposing we can still count the zero-interest-rate environment that has persisted for more than six years "abnormal", we have mostly shifted to a medium of exchange that pays interest, and the trajectory of technology (both information technology and financial technology) is toward more of that.
Traditional explanations of how monetary policy work often run more or less like this: the fed controls short-term interest rates, which affect the trade-off people make between holding their money in more liquid versus less liquid forms, and if they increase the amount they have in more liquid forms they spend more. As the most liquid form of money starts to pay interest at a rate that moves more or less one-to-one with other interest rates, we face something of a paradox; the interest rate is effectively zero in terms of the actual medium of exchange, and the "interest rate" that the fed targets simply measures the rate at which the value of the dollar declines relative to that.
If people at that point are largely using interest-bearing deposits and funds as the actual store of value and medium of exchange in the economy, to what extent does this "dollar" whose value declines relative to it even matter? At least at first, it can continue to serve as a unit of account. Indeed, at this point it seems to have retained that function in the United States, even as it has largely lost the others; even where you see contracts with "indexing" of some sort, it's far more often to a price index than to something connected to interest rates per se. Perhaps over time contracts could start to have future cash flows stipulated in terms of the amount of money that would be in a bank account at that point in time if a specified amount had been deposited at the beginning of the contract, but there's no logical reason why the new medium of exchange would need to take over this last function of money.
Thus the dollar, increasingly, serves only as a unit of account, and will maintain its relevance only if it continues to serve for many purposes as a better unit of account than some alternative. What makes a good unit account is not necessarily entirely the same thing that makes a good store of value or medium of exchange. To the extent that it does not, this new separation is in fact liberating for the Fed; it can focus on making the dollar a good unit of account, possibly allowing more volatility in its value than would be optimal if it were also a widespread store of value.
A business, for example, will typically have inputs that it purchases as it goes along, but will also require long-term inputs into the production process — a lease on a retail store, for example. (Employees who may, in principle, be freely dischargeable at-will employees, are probably in practice at least somewhat long-term inputs due to firm-specific knowledge and training and the costs of hiring and firing.) It also will produce products that may include short-term sales, longer-term contracts to supply clients, or both. It is likely that there will be some "duration mismatch" between inputs and outputs. In each case where the company is locked in to a decision years ahead of time, it risks a change in circumstances; if it is mostly selling as it goes along, it might wish to respond to an unexpected drop in demand by finding a way to cut production costs, but if it is selling mostly by long-term contract but has to buy its inputs day-to-day, it is subject to an increase in costs that it can't pass along. To the extent that it can specify prices in long-term contracts in terms of a unit of account that will drop in value if demand for its product goes down, or increase in value as competition for its supplies goes up, it will be easier for the company to responsibly engage in this business. A central bank that is trying to optimize its currency for use as a unit of account, therefore, will tend to devalue its currency when the economy in general is slowing down and increase its value (at least relative to expectations) when the economy is especially robust. These kinds of fluctuations in the value of actual holdings of the currency — long-term, as a store of value, or even short-term, as a medium of exchange, between the sale of one good or service and the purchase of another — will tend to make it less useful for those purposes. In a world where the central bank doesn't have to trade off these costs against the benefits of a countercyclical unit of account, it can focus on a better unit of account, while the other functions of money are provided elsewhere.
 There are (perhaps more compelling) arguments related to intertemporal substitution as well, but note that those explanations implicate the real interest rate rather than the nominal interest rate. You therefore need a story about how inflation and interest rates are simultaneously determined, and in particular why a decision by the fed to raise interest rates would reduce inflation expectations. These stories and explanations typically themselves come back to a "liquidity effect", so we're left with the same conundrum as the role of non-interest-bearing money atrophies.
To some extent, as long as the government is using it as a unit of account — specifying tax liabilities, contract payments, and social security benefits in dollars, and even taxing the deviation between our new electronic currency and the dollar as "interest income" — it can be kept relevant by fiat.