To answer the question in the title of this post, it's useful to think of an island with only two goods. One of the goods is non-renewable, but highly desirable. The other good is less preferred, but it is renewable (thinking of renewable and non-renewable energy resources, for example). The key is to distinguish between changes in prices that reflect changes in the relative scarcity of the two goods, and changes driven by increases in the money supply.
Over time, as the stock of the more desired good falls due to consumption, the price of this good will rise relative to the renewable good. Consumers will be hit by increases in the cost of living -- the same basket of the two goods purchased last year now costs more.
But is this the kind of increase in prices the Fed should respond to? No, the price increase -- and the increase in the cost of living -- reflects increasing scarcity of the desired good. The price of the two goods are changing to balance the relative supplies of the two goods. Unless the price of the non-renewable resource does not properly take account of the preferences of future generations -- and it may not -- or there is some other market failure, there is no reason for government to intervene to change the prices. If the prices are correct, they will allocate the resources optimally.
Now consider a different case. Suppose the central bank in charge of money -- sea shells of a particular type identified with the central bank's special mark -- and the money supply is being increased at a rapid rate. This will drive the prices of both goods up, but so long as the price of each good rises in proportion to the change in the money supply so that the relative price of the two goods is undisturbed, no problem. The price level will adjust to the number of sea shells in circulation, but since relative values remain intact, nothing will change.
However, suppose one of the two prices is sticky. It does not change very fast when the number of sea shells in circulation increases. In this case relative prices will be distorted as the number of sea shells increases, one price will rise faster than the other, and resources will be misallocated. In this case the Fed would want to do something about the inflation since it is having negative effects on the efficient allocation of the two resources. This is, essentially, the Fed's justification for activist policy.
A couple of notes. First, it's interesting to think about how technological change that improves the quality or lowers the price of the renewable good plays into this. Such a change could offset the increase in the cost of living that households face. Thus improving technology, not Fed policy, is the key to helping people on the island struggling with high prices.
Second, this is about the long-run and growth in demand. The central bank may still want to try to offset temporary price spikes, for example when sticky prices can cause problems that persist beyond the spike in the price of one of the two goods (e.g. a spike in the price of oil that leads to long-lived price distortions). But long-run growth that causes the price of one of the goods to rise by more than the other, i.e. relative price changes, is not something the Fed should try to neutralize.