Is there a reason not to show a student their letter of reference?
One important reason is if you make direct comparisons with other students, which is one of the most useful things you can do in a letter. If you say "this student is even more promising than X, and perhaps comparable with Y, but not quite as strong as Z", then it doesn't seem appropriate to tell the student. Even if your letter doesn't name anyone specific, but instead says "this student was among the top two in a class of twenty-five", it still leaks a little information about how other students performed in the class.
Even aside from direct comparisons, you'll be making implicit comparisons if several students compare the letters you wrote for them to see whose is more enthusiastic. That doesn't raise the same ethical issues, but it's still awkward. I have no interest in telling students my private opinion of how they compare with each other.
The second question is, is it reasonable to refuse to write a reference for a student when the student does not waive their right to see the reference.
Why not? The fundamental question here is whether you ever have an obligation to write a letter. In general I'd say no, although there are cases in which it would be offensive to refuse to write for anything less than very serious reasons. (For example, Ph.D. advisors owe letters to graduating students.) Under most circumstances, refusing to waive the right to see the letter sounds to me like a fine reason to decline to write one.
There are also all sorts of intermediate options you could choose. For example, you could tell the student you would have to write something like "I warned this student that I could not write a compelling letter if he did not waive the right to see it, because I would be unable to compare him with other students. He insisted that he needed a letter from me but was unwilling to waive this right, so I will do the best I can. I can confirm that he received an A in my class and has sufficient preparation for further study in this field."