Short answer is no. The skin actually helps protect the bulb.

Whoops! I did.

It will have to make another then, and it will, usually.

That sounds clever. They don't all come with skins anyway. Some have skins fully, others no skin at all, and some with just half their skin. And some have skin which has fungus on, but no fungus on the white bulb surface beneath, in which case it would seem best to remove from those.

The brown skins are just dead material but they do act as physical protection for the bulb. I do not worry about skins that have come away in the sales packaging but I would not remove the skin deliberatley.

Like Crocus, the skin is used as part of the diagnostic tools for identifcation of tulip species. The skin on the big hybrid types is not that important,but the more substantial covering of species types should be left alone.