Mr. Hooper wears the black veil because he has come to understand a certain truth about humanity: that we are all sinful, but we attempt to hide our sinfulness from one another by holding up a figurative veil between ourselves and everyone else.  This intangible veil separates us, for as long as we live, from our fellows because we can never truly be known or know another when we attempt to hide this crucial part of what makes us human.

The first sermon Mr. Hooper preaches after he dons the veil helps to make this clearer.  Its subject is "secret sin, and those sad mysteries which we hide from our nearest and dearest, and would fain conceal from our own consciousness, even forgetting that the Omniscient can detect them."  In other words, he speaks of our secret sinfulness and the need we feel to hide that truth about ourselves, even from the people we are closest to.  We would even prefer to forget this truth ourselves, and we can almost convince ourselves that even God is unaware of our secret sins because we are so anxious to conceal them.

What explanation does Mr. Hooper give Elizabeth as to why he refuses to remove his veil in "The Minister's Black Veil"?

When his fiancée Elizabeth asks Mr. Hooper to remove his veil because there may be "whispers" that he hides "under the consciousness of secret sin" rather than the "type of an innocent sorrow," as he claims, the minister smiles a sad, obscure smile and responds,

"There is an hour to come," said he, "when all of us shall cast aside our veils. Take it not amiss, beloved friend, if I wear this piece of crape till then."

Refusing to remove his mask until others remove their figurative masks and become honest about their sins, Mr. Hooper seems committed to getting people to acknowledge their humanity and their faults, a true challenge for Puritans for whom such exposure can lead to condemnation as they may be thought of as among the Unregenerate. No matter whatever his true intent is, Mr. Hooper clearly wishes to teach a moral lesson by wearing the black veil as a symbol that each man and woman can interpret according to their own consciences.
His act is certainly an honorable one as he is willing to be ostracized by members of the community who become uncomfortable when they cannot read what is on his own face and fear what he may know.

What explanation does Mr. Hooper give Elizabeth as to why he refuses to remove his veil in "The Minister's Black Veil"?

When Elizabeth, Mr. Hooper's fiancee, asks him to remove the veil and explain to her why he put it on in the first place, he responds that he cannot.  When he refuses to remove the veil, she presses him to at least explain himself, to "remove the mystery from his words".  The minister says only that he has vowed to wear the veil forever, as "a type and symbol", and as she receives his explanation, Elizabeth is suddenly stricken with a sense of the symbolic horror of his act.  By refusing to let go of the black veil, Mr. Hooper is in essence displaying his willingness to give up even the most precious of human relationships, a relationship of love.

Mr. Hooper's black veil is symbolic of the masks all men wear to hide their innermost secrets and desires.  Just as the veil has the power to destroy the sacred relationship he might have had with his fiancee, so do the masks that all men wear have the potential to ruin their relationships with those around them in their lives.

What explanation does Mr. Hooper give Elizabeth as to why he refuses to remove his veil in "The Minister's Black Veil"?

Elizabeth, who is not afraid of the veil at first, finally cannot tolerate it any longer and asks him to remove it. When he refuses, he says he has vowed to wear it forever, as a "type and symbol". He explains the veil could also serve as either a symbol of mourning or a symbol of sorrow for the secret sin that he's accused of hiding. Of course, there has been all kinds of gossip as to why he's wearing the veil, and he is saddened that the veil would have such an effect.

The veil is a symbol of the mask we all wear to hide our deepest, darkest secrets and desires. He accuses all men of hiding under their own "veils" from God and other men. He views this as a sin or at least a weakness of mankind. At the end, when Hooper is dying, he asks that the other not judge him until they have examined their own consciences and found themselves free of sin. I am reminded of the Biblical verse that says, "Let those who are without sin cast the first stone." Hooper feels he has admitted his sin openly by wearing the veil while others wear a veil on their souls. He says it is human nature for us to commit and then try to hide our sins.

Why does Mr. Hooper refuse to remove the veil?

One day, quite suddenly and unexpectedly, Mr. Hooper, a minister in a small village, begins wearing a black veil in front of his eyes. He refuses to remove it despite the shock and horror of his congregation. Most people are afraid to ask Mr. Hooper about the veil, but his beloved, Elizabeth, dares to do just that. Mr. Hooper, however, refuses to explain his new habit, even when Elizabeth says she can no longer be with him.

In fact, the only answers Mr. Hooper gives Elizabeth are as veiled as his face. He says he has taken a vow to wear the veil at all times for the rest of his life and that it is a symbol (although he does not say of what). He remarks that if the veil is a “sign of mourning,” then perhaps he has a dark sorrow for which he is doing penance (although he merely suggests this as a possibility; he does not say it is true). He also says that if the veil is covering a secret sin, then everyone has the same reason to cover their face. Yet he never openly says why he refuses to remove his veil.

As the years go by, Mr. Hooper is both venerated and feared. He is an excellent minister, yet people are terrified of him. Children run in the opposite direction when he approaches, and he remains mostly in solitude throughout his life. However, he never removes his veil.

Even on his deathbed, Mr. Hooper keeps the veil over his face. In his last words, he provides a solid clue for his actions over all these years. “Tremble also at each other,” he commands. People have avoided him, shown him no mercy, and even fled from his presence because of that black veil. He has revealed to them his “inmost heart,” but they cannot see beyond the veil, which he calls “the symbol beneath which I have lived and died.”

The veil, then, appears to be a test and a lesson to others. It shows them how much they judge by appearances alone rather than looking beneath the exterior to the person within. With his very life and a simple black veil, Mr. Hooper has revealed the shallowness of his neighbors, their lack of love, and their inability to see beyond the veil, literally and figuratively.

In "The Minister's Black Veil," why do the villagers bury Mr. Hooper without removing the veil?

When the Reverend Mr. Clark sits by Father Hooper's deathbed, Mr. Clark asks Father Hooper to allow him to remove the veil from Father Hooper's face so that he can meet eternity without it.  Father Hooper, however, is horrified by this idea, and he shrieks, "'Never! [...].  On earth, never!'"  It is clear that Father Hooper wishes never to have the veil removed, and this may be one reason that his parishioners do not remove it.

Further, everyone gathered at Father Hooper's deathbed still seems to fear the mysterious veil.  Mr. Clark even suggests that it signifies some terrible sin Father Hooper had committed.  However, Father Hooper asks, 

"Why do you tremble at me alone? [....] Tremble also at each other!  Have men avoided me, and women shown no pity, and children screamed and fled, only for my black veil?  What, but the mystery which it obscurely typifies, has made this piece of crape so awful?  When the friend shows his inmost heart to his friend; the lover to his best beloved; when man does not vainly shrink from the eye of his Creator, loathsomely treasuring up the secret of his sin; then deem me a monster, for the symbol beneath which I have lived, and die!  I look around me, and, lo! on every visage a Black Veil!"

It seems to be, in part, this "mystery" that Father Hooper describes that so terrifies the people.  They seem always to have had an obscure idea of what the veil symbolizes -- that each of us has secret sins that we attempt to hide from each other, ourselves, and even God -- but no one wants to admit that they might understand because that would be tantamount to admitting that one has these secret sins on one's soul, and what everyone wants the most is to hide this very fact.  If, in truth, these people really do have even a vague understanding of the veil's meaning, then they would not want to remove it after Father Hooper has died because it is a meaningful and accurate symbol; if, on the other hand, they really don't have a concept of the veil's meaning, then they would not want to remove it because it is such a mystery, and we fear mysteries.

Why doesn't Mr. Hooper remove the veil before he dies in "The Minister's Black Veil"?

Good question.  Look at his line late in the story: "Never!" cried the veiled clergyman. "On earth, never!"

From some reason—some deep and mysterious reason—the veil has become an intense symbol for the minister. He's essentially sworn an oath to himself never to remove it, and thus to never stand bare-faced before the world, while he lives. It's taken on a spiritual significance; this is a veil that is also the veil of tears through which all mortal men must see the world. In a sense, he can't; he's admitted he's sinful, and so he must wear this as long as he breathes.

Why might Mr. Hooper have put on the black veil in the story "The Minister's Black Veil"?

Your question correctly poses the conditional "might" as a venue to explain the potential reasons why Hooper makes the odd decision to wear a black veil in front of his face for the rest of his life. 

In reality, this is the one question in the story that never gets answered. The reactions that the veil elicits in the people of the village, however, can somewhat help us to conclude that Hooper purposely wanted to cause specific emotions among the people that reflect what lurks within their hearts. The ultimate reason why he would want this will still remain a mystery that the reader will have to deduce. 

A possible clue to help form a conclusion can be drawn from the minister's answers to his fiancée, Elizabeth, when she demands to know what is going on. Since they are engaged to be married, Elizabeth feels that she has a right to know what could be possibly driving her future husband to make a choice of this nature. However, Hooper's answers are problematic because they are not final. At one point he says to her that, if his veil were a symbol of mourning, 

"I perhaps, like most other mortals, have sorrows dark enough to be typified by a black veil." "If I hide my face for sorrow, there is cause enough . . . and if I cover it for secret sin, what mortal might not do the same?"

If he is not willing to tell the woman that he intends to marry about his rationale, why would he tell anyone else?

Therefore, we can only speculate that the minister wore the veil to cause in the parishioners every possible feeling of uneasiness. Only by understanding exactly what causes their emotions could they be able to learn the inner fears and anxieties that lead them to feel them. This would be a lesson like no other that the minister would be teaching his flock. This also "might" have been the reason behind his decision. The best evidence for this theory is found at the end of the story, when Hooper confronts those visiting his deathbed and the emotions that he conjured in them through the simple fact of wearing a veil. 

"Why do you tremble at me alone? [...] Tremble also at each other! Have men avoided me, and women shown no pity, and children screamed and fled, only for my black veil?"

Here he hits directly on how their inner fears are mirrored by the veil; hence, some equally dark and weird situations must be going on in their hearts. 

"What, but the mystery which it obscurely typifies, has made this piece of crape so awful? [...] When man does not vainly shrink from the eye of his Creator, loathsomely treasuring up the secret of his sin; then deem me a monster, for the symbol beneath which I have lived, and die! I look around me, and, lo! on every visage a Black Veil!"

With this, he directly states that he sees a veil in everyone, part of the masking and deception so typical to humans. Why fear the man who dares to expose the lies of the soul? This is a lesson he might have wanted to teach.