We hesitate to describe this as an open-and-shut case, but the quick answer is No, shut has not replaced closed. In fact, ever since it first appeared in English more than 800 years ago, closed has proven itself an impressively serviceable adjective.

In addition to meaning "not open," closed can be synonymous with "enclosed" (as in closed porch), and it can be used in a number of mathematical applications, ranging from closed set to closed curve to closed interval. Closed ballroom dancers join their bodies together, a closed universe has enough mass to stop expanding and eventually collapse, and a closed racetrack has the same starting and finishing point. All in all, we counted more than a dozen senses of closed.

And shut? That adjective is almost as old as closed but it has many fewer senses: two, to be exact. Shut means "closed, fastened, or folded together" (as in, the shut door was blank against the winter sun); and, when paired with "of," shut means "rid, clear, or free" (as in, the year-round residents were finally shut of the summer visitors).