Is it ever mentioned in the movies or the books whether it is possible to vanquish or get rid of a ghost forever?
I am not talking about a ghost somehow moves on by itself, more like an evil ghost being handled by living wizards somehow.
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But it does not appear to be possible.
Besides the non-death of Nicholas de Mimsy-Porpington by the Basilisk mentioned in the other answers (and the nice speculation about moving on), the best evidence I can think of is statements to the fact that Peeves cannot be removed by any known magic (which would presumably include “killing” him, something that would not likely be considered immoral against an already dead being).
Becoming a ghost involves premeditated magic manipulating one's soul, presumably doing something to it that either tethers it to this world or untethers it from the natural universe. (Differing from the dark kind of magic used when, for example, tethering a piece of a soul to an object when making a horcrux.)
If that is the case then perhaps it is possible to do something to undo that modification; but again with reference to Peeves, that something does not appear possible by beings other than the willing ghost himself or herself.
 Unless JKR has specifically stated otherwise, of which I am unaware.
 I am not convinced that ghosts in HP are really the same kind of sentience that living witches and wizards are; IIRC they are described as more of a memory frozen in time. (See Ghosts - Pottermore.) I would personally classify it as “dark magic” because it interferes with the witch or wizard’s natural being and status, but the Wizarding World seems to want to ignore it as an issue.
We see in Harry Potter and the Chamber of Secrets that, even when the Basilisk stares straight at Nearly-Headless Nick, he doesn't "die" in the sense that your question is asking.
The basilisk kills people by looking at them... Justin must’ve seen the basilisk through Nearly Headless Nick! Nick got the full blast of it, but he couldn’t die again...
Harry Potter -- Harry Potter and the Chamber of Secrets
So we see that even such a Dark and dangerous creature as a basilisk cannot kill a ghost when they're already dead.
However, also mentioned in the book is that Nearly-Headless Nick was blackened and became completely disabled, Petrified. Possibly, this could be a desired effect for the wizards looking to disable a ghost! They just wouldn't Unpetrify them.
This isn't really killing or destroying them. But Nearly Headless Nick was petrified by the basilisk.
It was Nearly Headless Nick, no longer pearly-white and transparent, but black and smokey, floating immobile and horizontal, six inches off the floor.
Harry Potter and the Chamber of Secrets, Chapter Eleven (The Dueling Club)
He was completely inert and McGonagall was able to use a fan to blow him to the hospital.