What does Mr. Hooper see on everyone's faces on his death bed in "The Minister's Black Veil" and why is it important?

As Mr. Hooper is dying in his bed near the end of the story, another minister attempts to remove Mr. Hooper's black veil. He wants people to see Hooper's "triumphant" face as he passes over into eternal life. He also wants whatever Hooper may be hiding by the veil to be lifted from him before passing on to God's judgment. Hooper forcefully grabs the veil and says that he will never remove the veil "on earth." Then in a moment of great strength, Hooper addresses the people in the room around him, and he tells each of them that he sees a black veil on each of their faces.

I look around me, and, lo! on every visage a Black Veil!

This is important to the overall meaning of the story because readers are meant to interpret the veil as a concrete symbol of a person's hidden sins. People sin, and they try to hide it from God and friends alike. Hooper's veil is physically expressing this concept of people trying to hide the sin in themselves. Hooper never comes out and expressly tells people this is what the veil is for; however, on his deathbed, he finally announces to those around him what the veil is symbolic of. The revelation hits the people like a hammer blow because they immediately understand and even fearfully "shrank" away from each other. Hooper knows they finally understand, and he lets a rare smile be seen.

While his auditors shrank from one another, in mutual affright, Father Hooper fell back upon his pillow, a veiled corpse, with a faint smile lingering on the lips.

What foreshadows Mr. Hooper's deathbed scene in "The Minister's Black Veil"?

The short story "The Minister's Black Veil" by Nathaniel Hawthorne tells of a minister, Mr. Hooper, who one day appears at the meeting house with a black veil draped over the top half of his face, obscuring everything from his mouth up. His congregation is shocked. His fiancé pleads with him to remove the veil, but he refuses. As a result, his marriage is called off, and he is ostracized from the company of others. He does, however, continue to be an attentive, loving, and efficient pastor his whole life until he is old and on his death bed. As he is dying, the minister attending him implores him one last time to remove the veil, but Hooper declares that he will never do it in this life. He also affirms that on everyone's heart is a black veil, as if the veil covering his face is a metaphor for the human condition.

Foreshadowing is a literary device through which the writer gives clues about what will happen later in the story. In "The Minister's Black Veil," Hawthorne uses foreshadowing at important moments to indicate what inevitably happens during the deathbed scene.

The first incident of foreshadowing in the story is at the funeral of the young woman that Hooper officiates. As he leans over to bid farewell to the deceased woman, his veil momentarily leaves his face, and someone watching says that the corpse shudders. When Hooper delivers the prayer, he expresses hope that those attending will be ready for "the dreadful hour that should snatch the veil from their faces." In this he implies that they all wear metaphorical veils and only death can remove them.

Later, the foreshadowing is more explicit when Hooper talks with Elizabeth, his "plighted wife." She asks him to put it aside as they speak. He replies that to him it is a symbol, and he will not remove it in this life.

Know, then, this veil is a type and a symbol, and I am bound to wear it ever, both in light and darkness, in solitude and before the gaze of multitudes, and as with strangers, so with my familiar friends. No mortal eye will see it withdrawn.

We see, then, that there is foreshadowing in the story not only of Hooper's insistence that he will wear the veil to his death but also of his deathbed declaration that his veil is a symbol of the veil that everyone has over their hearts.

Mr. Hooper's deathbed scene is foreshadowed by his conversation with Elizabeth, the woman who he was supposed to marry, at the beginning of the story.  When she approaches him to ask directly about the veil -- something no one else in the town has been brave enough to do -- she requests, first, that he remove it.  He refuses, saying, "'No mortal eye will see it withdrawn.  This dismal shade must separate me from the world: even you, Elizabeth, can never come behind it!'"  He tells her clearly that he will never take the veil off during his lifetime; therefore, when Mr. Clark asks him to remove it in the moments before his death, it should be no surprise that Mr. Hooper has a dramatic and emotional response.  "Father Hooper snatched both his hands from beneath the bedclothes, and pressed them strongly on the black veil, resolute to struggle, if the minister of Westbury would contend with a dying man."  Father Hooper cries out "'Never! [...] On earth, never!'" 

Further, when speaking with Elizabeth, Mr. Hooper had asked, "'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?'" He thus essentially reveals to her that the veil is a material representation of the fact that we all try to hide our secret sins from one another.  And yet, he lives his entire life being treated as an outcast because people seem to think that he, alone, as committed and hides some secret sin.  However, on his deathbed, Father Hooper cries, "'Why do you tremble at me alone? [...].  Tremble also at each other! [....] I look around me, and, lo! on every visage a Black Veil!'"  He again indicates that it is not only he that possesses these secret sins, but everyone, the same thing that he told Elizabeth many years prior.