Why did Elizabeth say he sould remove the veil?
In reality the reason is quite simple: He was the man she was to marry, and everyone thought he was insane. She had to bear the burden of being associated with him, so she implored him more than once to remove it so that people can actually see he is a normal person of good character. She was also in the hopes that his removing his veil would make a stronger point in case than attracting attention so negatively. In the end it did not matter, because he still refused to remove it.
In the short story "The Minister's Black Veil" his wearing of the veil has made everyone in the parish and community uncomfortable. By seeing the veil they are forced to recognize that they carry their own hidden sins within. Elizabeth could have been uncomfortable with the veil for several reasons.
1) She could have thought it was just ridiculous.
2) She could also have felt uncomfortable as it made her reflect on her own sins.
3) She wanted to be able to see her husband's face because socially it is much more comfortable to be able to look at someone one is talking with rather than a piece of cloth.