We know (from the books) the process of applying a new face involves drinking a potion, having your real face cut, and applying the new face. The wearer of the new face cannot tell the difference, but all features including scars and teeth are transferred. The wearer may also experience visions of the person the face once belonged to.
I would assume that since the application of the faces involves magic, their preservation involves magic as well. The faces seem to be "revived" by the blood of their new wearer.
Below is a long transcript of "a girl" (aka Arya) going through her first face change into the "ugly little girl". While this scene does not make into the show, it is our most in-depth description of the process.
Still as stone, she thought. She sat unmoving. The cut was quick, the blade sharp. By rights the metal should have been cold against her flesh, but it felt warm instead. She could feel the blood washing down her face, a rippling red curtain falling across her brow and cheeks and chin, and she understood why the priest had made her close her eyes. When it reached her lips the taste was salt and copper. She licked at it and shivered.
"Bring me the face," said the kindly man. The waif made no answer, but she could hear her slippers whispering over the stone floor. To the girl he said, "Drink this," and pressed a cup into her hand. She drank it down at once. It was very tart, like biting into a lemon. A thousand years ago, she had known a girl who loved lemon cakes. No, that was not me, that was only Arya.
"Mummers change their faces with artifice," the kindly man was saying, "and sorcerers use glamors, weaving light and shadow and desire to make illusions that trick the eye. These arts you shall learn, but what we do here goes deeper. Wise men can see through artifice, and glamors dissolve before sharp eyes, but the face you are about to don will be as true and solid as that face you were born with. Keep your eyes closed." She felt his fingers brushing back her hair. "Stay still. This will feel queer. You may be dizzy, but you must not move."
Then came a tug and a soft rustling as the new face was pulled down over the old. The leather scraped across her brow, dry and stiff, but as her blood soaked into it, it softened and turned supple. Her cheeks grew warm, flushed. She could feel her heart fluttering beneath her breast, and for one long moment she could not catch her breath. Hands closed around her throat, hard as stone, choking her. Her own hands shot up to claw at the arms of her attacker, but there was no one there. A terrible sense of fear filled her, and she heard a noise, a hideous crunching noise, accompanied by blinding pain. A face floated in front of her, fat, bearded, brutal, his mouth twisted with rage. She heard the priest say, "Breathe, child. Breathe out the fear. Shake off the shadows. He is dead. She is dead. Her pain is gone. Breathe."
The girl took a deep shuddering breath, and realized it was true. No one was choking her, no one was hitting her. Even so, her hand was shaking as she raised it to her face. Flakes of dried blood crumbled at the touch of her fingertips, black in the lantern light. She felt her cheeks, touched her eyes, traced the line of her jaw. "My face is still the same."
"Is it? Are you certain?"
Was she certain? She had not felt any change, but maybe it was not something you could feel. She swept a hand down across her face from top to bottom, as she had once seen Jaqen H'ghar do, back at Harrenhal. When he did it, his whole face had rippled and changed. When she did it, nothing happened.
"It feels the same."
"To you," said the priest. "It does not look the same."
"To other eyes, your nose and jaw are broken," said the waif. "One side of your face is caved in where your cheekbone shattered, and half your teeth are missing."
She probed around inside her mouth with her tongue, but found no holes or broken teeth. Sorcery, she thought. I have a new face. An ugly, broken face.
A Dance with Dragons - The Ugly Little Girl