Apparently happens in Ao No Exorcist. Father Fujimoto is definitely an example. So is Toudou, later on, although it does seem to depend on how powerful the demon in question is.
Animals possessed by spirits in Berserk get deformed human-like faces
In Bleach, when Kaien is possessed by Metastasia he first gets hollowed, empty eyesockets, then he starts sprouting tentacles from his limbs, sockets and mouth.
In Soul Eater, when Medusa loses her body, she possesses a young girl and twists her appearance into a younger version of herself.
Happens twice in Constantine. In the beginning, when the soldier demon inhabits the young woman her face is covered with blotches. When Angela Dodson is possessed by Mammon (Lucifer's son) near the end the change is much greater. In each case, once the possession ends the skin changes disappear.
The Exorcist had a pretty gruesome transformation.
Legion: various angels do odd things to the bodies they inhabit.
The Devil Inside plays it straight, but in fairly plausible ways. Those possessed sport freakish flexibility, dilated eyes (a line of dialogue indicates that the human eye can dilate to 9 mm, 11 mm under the influence of some drugs, but possession can force it further), and red eyes. All could be explained by actual physical symptoms.
The Candarian demons in the Evil Dead series use this in a variety of ways.
In the original film the only change was in the eyes (and spreading black veins in one case), but physical damage inflicted on the host quickly reached gruesome levels.
Evil Dead 2 had a greater degree of Body Horror, including a possession victim with an extremely long neck.
Army of Darkness had Sheila suffer some minor changes in bone structure when she was possessed, and the rest of the Deadites took on a rotted appearance even without any injuries.
Finally, the 2013 remake only gives the possessed yellow eyes by default, but they then inflict gruesome injuries on themselves, apparently to match the documented appearance of whatever spirit is possessing them.
Literature
True for most Denarians in The Dresden Files, though some of them choose to keep their human appearance.
Played With in Perelandra, Dr. Weston's possessed body at first looks quite normal but as it is damaged beyond repair in the course of the plot (yet continues to move thanks to the devil inside it), it becomes horrible to look at and wholly inhuman.
In Inheritance Cycle, when a human is possessed by spirits and becomes a Shade, it transforms their appearance, including turning their hair bright crimson.
Spellbent: Possession victims get funky eye colors. Their sclera (white of the eyes) turns red or in some cases purple, depending on the power level of what's possessing them.
In Kamen Rider Den-O being possessed by any of the friendly —taro imagin gives the character a lock of appropriately colored hair and sometimes a change in wardrobe.
Dungeons & Dragons 3.5 & 4.0: Many fiends find remodeling their new body to their advantage and extensive remodeling can result in being completely replaced.
In Warhammer40000, Daemons will alter the body of their host, moulding it into one more suitable for them. The host rarely lasts for very long.
In Demon: The Fallen, Demons knowledgeable in the lore of transformation can make permanent alterations to its host.
The Fomori in Werewolf The Apocalypse. When a bane (an evil spirit of the Wyrm) possesses a human body, that person's body is hideously mutated with whatever the Storyteller picks from a grab-bag of disgusting Body Horror-based powers. Examples include multiple limbs and mouths, corrosive puss boils, and internal organs replaced by infectious fungus.
Videogames
In Diablo 2, this happens to the first game's protagonist when he fails to contain Diablo's soul.
In Doom 3 (and, quite possibly, the old Doom games) demonic possession turns people into zombies.
This happens to anyone possessed by a demon in Dragon Age.
Legacy of Kain: Raziel learns to occupy corpses and shape them into an image of himself.
When an Ing from Metroid Prime 2 fuses itself to a host, the host takes on a darker, mutated appearance as a direct result.
Super Robot Wars UX: Noval Dilan becomes the physical embodiment of Kali Yuga (with Durga's appearance) and grows to the size of a planet.
Emily from Our Little Adventure took on a horrid alien appearance as some kind of controlling, Angelo loyal parasite inside her took control of her body.
Web Originals
Noted in the Whateley Universe as one possible consequence of a low-powered Avatar-type mutant hosting a too-powerful spirit as the spirit will make changes to the Avatar's body to make itself more comfortable. And the Avatar trait is already all about being superhumanly good at this sort of thing...
Western Animation
After being possessed by Zordrak in an episode of The Dreamstone, Amberley gains a pair of extremely evil looking glowing eyes.
Jackie Chan Adventures: the ghost of Shendu (a dragon-demon-wizard) usually gives his host glowing red eyes when in control. While he's not capable of transforming his host's body under his own power, he once tried to do so manually with potions.
In Adventure Time, the Ice Crown causes a physical transformation in its wearer, whom it tries to impose its own will on. As the person falls further and further under the Crown's control, they begin to develop blue skin, long white hair, and fangs (although some of these traits can vary depending on circumstance).
