For "reasonable" cutting speeds (less than, say, the speed of sound in the material) the total amount of work required to part the material (as with a blade) or chop a small amount of it into shreds (as with a saw having teeth) does not depend on how fast you are performing the work.
However, for very fast cutting speeds, there is no time for the frictional heat generated by the cutting process to be dissipated by the cutting blade(s) or the material being cut. This heat buildup will eventually degrade the strength of the cutter by a variety of mechanisms, causing it to quickly become blunted.
In general, then, faster cutting speeds can cause abnormally short tool life unless the cutter is specifically designed to maintain its strength at high temperatures. A perfect example is tool steel, also called "high speed steel" or HSS, which can be used to cut metal at such a speed that the tool tip is red-hot from the friction.