I know the palantír can be used to see afar or communicate with other palantíri, but is there a device, character, or magical ability that allows one character to possess the consciousness and/or body of another being?
I can think of no full possession of a living creature. The closest I can think of, aside from (as mentioned in the comments above) the One Ring's influence over its bearer and Saruman and Grima's influence on Theoden (and similarly, Sauron's apparent poisoning of Denethor's thoughts):
Sauron had strong influence over the bearers of the nine rings of Men (more subtle at first, but ultimately turning them into the Nazgûl), and some corrupting influence through the seven Dwarf rings.
The Barrow Wights could take control of interred remains, and apparently cause paralysis among the living.
Dragons like Glaurung could hypnotize people.
In general, Maiar Saruman and Sauron (as Annatar) had the power to beguile with words.
The Vala Irmo apparently had some sort of power over dreams.
There's no example that exactly matches your description. A couple of somewhat similar examples:
First it should be noted that the scene in the LotR movie where Saruman takes control over Théoden and sees through the King's eyes isn't canon, nor is the scene where Gandalf and Saurman is fighting over the control of Théoden.
It the books, some manner of power was used to make Théoden complacent, weak and aged, but it seemed that he was mostly affected by Wormtongue's manipulations more so than something supernatural.
Similar to the above, Sauron manages to break Denethor's will by communicating with him through the palantír. He doesn't directly control Denethor, but breaks his will and causes him to despair and turn mad.
There is one example where the Sindarin Princess Lúthien uses some form of magic to let her and Beren take full possession of two dead, defeated beasts of Morgoth: a monstrous bat (Thuringwethil) and and a wolf (Draugluin).
The exact nature of the spell she used isn't described, only that the disguise was good enough to let them enter Angband itself. It could have been an illusion that made them simply look like the creatures, or it could be something that actually let them possess the dead bodies. Regardless, it is clear that the creatures were dead when the spell was cast.
Another example is the dragon Glaurung using some manner of mind control to manipulate Túrin, paralyzing him and making him believe that there was a chance for him to save his family members, instead of saving the war prisoners taken from Nagothrond. But Glaurung doesn't control him directly. Glaurung also uses his hypnotic powers to make Nienor lose all memory of her previous life.