Can holding interest rates at zero (ZIRP) for a very long time cause the deflation it is trying to prevent? The fisher equation is nominal interest rate = real interest rate + inflation rate.

If nominal interest rate is kept to zero, and real interest rate remains constant positive in the long term, then the inflation rate will be negative, representing deflation!

When the central bank drops nominal interest rate at 0%, the negative delta in the nominal interest rate increases money supply and increases the price level - for a period of time. The boost in demand is because demand is brought forward due to the fall in interest rates. Producers increases production to match this surge in demand, and in doing so, increasing supply of goods come into the market. If there are no more negative deltas in interest rate coming, as is the case under a zero interest rate policy where the nominal interest rate is zero bound, the increased supply is going to cause a decrease in price level due to decreased demand that was consumed to produce the earlier demand surge from the negative interest rate delta, causing deflation.

I'm sure there is a flaw to the logic above, and to the fisher equation, otherwise it would mean a ZIRP, held for a long time, would produce the deflation it was implemented to prevent! Can you point it out?
While my answer would depend on how, why, and what the central bank said about why they pushed nominal interest rates to zero, I suspect in most situations the answer is no.

Consider the simplest case of quantitative easing. Continuously injects cash into the economy and uses it to purchase public debt and if that is exhausted private debts. With this massive increase in the money supply, and the expectation that the money supply would continue to increase in the future, it would be unlikely that prices would be falling (deflation) in such a situation. 

Why? Recall Milton Friedman's quip that "Inflation is always and everywhere a monetary phenomenon." While it may not be an iron law, there does seem to be good evidence that when the money supply increases prices are likely to go up after. 


  Is inflation always a monetary phenomenon? Many economists believe
  that the link between money growth and inflation in the U.S. has
  weakened over the last two decades due in part to the Federal
  Reserve's policy experiment in 1979–1982 and innovations in the
  financial sector of the economy. I find that the long-run relationship
  between money growth and inflation is strong in a statistical sense
  and important economically. The key result is that the trend or growth
  component in CPI inflation is entirely due to the trend component of
  monetary base growth.


THE LONG-RUN LINK BETWEEN MONEY GROWTH AND INFLATION (CROWDER (2007)) 

Since most zero interest rate policies are likely to hold rates down by increasing the money supply, it is unlikely they will be deflationary. 

My major caveat is that a zero interest policy may be accompanied by overt or implicit indication that the central bank has a inflation target that is near zero. If the central bank tightens rates whenever they go above 0 percent they may have zero or even negative average nominal rates and deflation. But in that situation it it is the expectations of low future inflation and aggressive inflation fighting that hold down prices and the low inflation holds down the interest rates. Scott Sumner and others have argued that this was essentially the Japanese central bank's policy for many years.