If we are talking D&D, Druids may use metal swords to stab people and metal knives to cut small game. They have no problem with properly forged metals in these cases, so I wonder why a razor would be taboo. (Druids are also fine with haircuts, last time I checked.) The general goal of Druids were not complete destruction of all technology, but a balance or equilibrium between civilization and wilderness.

That said, as Slipperychicken mentioned, there might be very good RP reasons (perhaps even ingrained into druid society) that would make it taboo or frowned upon. And individuals may decide not to do so for personal reasons, of course.