It would seem logical that this would result in reverse-inflation or lower prices, but this is actually not the case (at least over time).
The banking system has a "supply and demand" market for reserves. When say 1 million dollar is withdrawn and not redeposited, the banking system in the aggregate will bid more for current reserves to replace the ones that were lost. The Fed does not like to see changes in the rates the bankers lend each other reserves and if it thinks this exchange rate is higher than the targeted Fed Funds rate, it will sell REPOs to primary dealers in an attempt to move this rate down.
So in the end, the amount of money will roughly be the same (as near as I can tell). What will change is that the Fed will have new assets it didn't have before. These assets in turn will accrue interest, and profits from the Fed get turned over to congress. So in a round about way, burning money is a transfer of wealth from the individual to the public.
In the same way, counterfeiting does not actually increase the money supply in the US over time. New fake dollars are deposited, drive down reserve rates which results in the Fed selling assets to absorb the new money. Less Fed assets = less taxpayer money. So counterfeiting hurts, but just not in the way most suspect.