There are two respects in which standard theory tells us the zero lower bound on nominal interest rates matters. First, when short-term nominal interest rates are zero, conventional open market operations do not matter - there is a liquidity trap. If the central bank swaps zero-interest-rate reserves for zero-interest-rate Treasury bills, this should be irrelevant, in that no prices or quantities change. Further, a liquidity trap is also a feature of regimes like the one we currently have in the United States. When there is positive supply of excess reserves in the financial system, and reserves bear interest, then the interest rate on reserves (IROR) is determining all short-term interest rates. If the central bank swaps reserves for Treasury bills, that will also be irrelevant under current circumstances.
Of course (again, under current circumstances), if the central bank lowers the IROR, this will not be irrelevant. The problem for the Fed is that, with the IROR at 0.25%, there is little room to move. As Ben Bernanke stated in his Jackson Hole speech, a decrease of the IROR to zero would have little effect and "could disrupt some key financial markets and institutions." It certainly seems correct to say that lowering the IROR to zero would have little effect, but arguing that this would be disruptive seems wrong. In any case, this was part of the argument for QE2, the recently-embarked-upon purchase of $600 billion in long-term Treasury securities by the Fed, under the premise that something needed to be done, and there was no other avenue to pursue.
Second, the zero lower bound on short-term nominal interest rates plays an important role in New Keynesian analysis. In a typical New Keynesian sticky-price model (e.g. see Woodford's book) the nominal interest rate is the policy instrument, and in some versions of these models it can be manipulated to achieve efficiency. Under the appropriate monetary policy rule, the "output gap" can be closed, and the flexible-price equilibrium is recovered, which is the equilibrium of the underlying real-business-cycle model. However, the zero lower bound can get in the way. There can be circumstances where the real interest rate is too high, relative to what it would otherwise be with flexible prices, but the nominal interest rate is at the zero lower bound, so that monetary policy is powerless, in terms of closing the output gap. This is basically the idea in this paper by Eggertsson and Woodford.
Now, apparently some central banks in the world are not constrained by the zero lower bound in the way that the Fed appears to be. Indeed, as you can see here, the Swedish central bank set its "deposit rate," its version of the IROR, at -0.25%, from July 2009 to September 2010. If this had been possible in the United States, this would seem like a simple solution to the Fed's current policy problem. There seems agreement on the FOMC for expansionary policy, and a straightforward way to implement this, rather than engaging in a massive experiment in long-maturity Treasury purchases, would have been to charge financial institutions for the privilege of holding reserves. Is this even feasible? Well, probably not. If you follow the links from here, you will not find anything to indicate that negative interest on reserves is legal in the United States. Congress gave the Fed the power to pay interest on reserves held by depository institutions. This legislation came with at least two flaws. First, the power to change the IROR was given to the Board of Governors rather than the FOMC, where such power would appropriately reside, given the importance of the IROR as a policy tool. Second, the legislation did not permit the payment of interest on reserves held by GSEs (i.e. Fannie Mae and Freddie Mac), which in practice has created a gap between the IROR and the fed funds rate. Add to these flaws that the legislation does not permit a negative IROR. Presumably this is not allowed as it would be interpreted as a tax. So much for foresight.
Given that the IROR cannot be negative, does this mean we are stuck with the zero lower bound, and the relative price distortions that are the concern of New Keynesians? Well, no. As discussed in this speech by Narayana Kocherlakota and in this paper and this one, fiscal policy gives us a lot of flexibility. Basically, in New Keynesian models, the problems created by sticky prices are relative price distortions which lead to a misallocation of resources. What could be more natural than correcting such distortions with fiscal policy, given a sufficiently rich array of taxes? The papers I link to above show that efficiency can be achieved in sticky price environments solely with appropriate taxation. Further, there are inventive ways to manipulate tax rates so that the zero lower bound no longer matters. Unless we think that fiscal policy is somehow constrained (e.g. the legislative process is too awkward) relative to monetary policy, this casts some doubt on the New Keynesian approach to monetary policy, as set out in Woodford's "Interest and Prices." The sticky price problem, if it exists, appears to be a problem more appropriately addressed with fiscal policy instruments than monetary policy instruments. Further,the solution is not an Old Keynesian program of expansion in government purchases of goods and services, but could be a revenue-neutral change in taxes.