In Harry Potter, we know ghosts became ghosts when they are too afraid of death to go on - I thought about it and I'm quite certain they all go to their own version of Harry's King's Cross and become ghosts when they return as Harry did. So can they return there? Once they've been a ghost long enough that they no longer fear death, can they move on?
As Binns became a ghost, he can no longer move beyond the veil, and will remain earthbound for all eternity. This means that unless he resigned or was sacked, he would be the History of Magic teacher forever. This is probably the reason that the school has not sacked him thus far, despite his dreadful teaching technique.
This statement is repeated, with the same meaning but not as long in the article on Binns. They do not, however, cite a source for this -- whether it was in books or Word of God.
The same information is given on the wiki article on ghosts:
As ghosts are imprints of souls of the said deceased wizards and witches, they are unable to move beyond the veil, and are forever trapped in the world of living as intangible beings. Due to this, few would choose this path. Depending on the conditions of the person's soul, they may not be able to return, as Tom Riddle's heavily damaged soul is trapped in limbo for eternity and cannot become a ghost.
Again, no citation in the wiki for the actual source.
Recall what Moaning Myrtle says in CS chapter 16.
‘My whole body sort of seized up, and then I was floating away…’ […] ‘And then I came back again. I was determined to haunt Olive Hornby, you see. Oh, she was sorry she'd ever laughed at my glasses.’
If haunting Olive Hornby was the reason Myrtle returned as a ghost, then that purpose is served now. Myrtle remains in Hogwarts whereas Hornby isn't there, so it's very likely that Olive has either died or Myrtle has given up on haunting her.
In any case, Myrtle seems to have no reason to remain in this world anymore. She seems to be generally unhappy and we see her do nothing but dwell her time, living in the bathroom and occasionally spying on boys in other bathrooms (see GF chapter 25). If she knew a way to go on, I'm sure she would.
According to JKR, Ghosts are transparent, moving, talking and thinking versions of wizards and witches who wished, for whatever reason, to remain on earth. (TOBTB - page 79 - UK Edition). I think it's a little different for ghosts than it was for Harry. Harry did not fear death -- in fact he braved it willingly. Nearly Headless Nick explains that ghosts are people who fear death:
‘I was afraid of death,’ said Nick softly. ‘I chose to remain behind. I sometimes wonder whether I oughtn’t to have ... well, that is neither here nor there ... in fact, I am neither here nor there ...’ He gave a small sad chuckle. ‘I know nothing of the secrets of death, Harry, for I chose my feeble imitation of life instead. I believe learned wizards study the matter in the Department of Mysteries –’
‘Don’t talk to me about that place!’ said Harry fiercely.
‘I am sorry not to have been more help,’ said Nick gently. ‘Well ... well, do excuse me ... the feast, you know ...’
Order of the Phoenix - chapter 38 - The Second War Begins
I can't find an instance in canon that indicates ghosts can move forward once they've died and come back, nor could I find a quote from JKR indicating this. A quote may well exist, but I wasn't able to find it (albeit in a fairly cursory search). Perhaps someone else will better be able to provide a quote, but as it is right now I don't believe there is canon evidence to support the idea that ghosts can move on. Some souls get trapped in limbo, such as Voldemort's, neither able to go back (like Harry did) or move on.
I personally feel that for a ghost to move on it would require something unethical. To have remained behind after death, I think they would have had to sever their souls from their bodies while alive, and that to die they would have to take over someone else's body and die in that. Hence why none seem to have done it.
Ghosts cannot move on. When they enter limbo they're faced with both the oppertunity to move on to a land that no one knows or to return to earth for certain reasons like unfinished business e.g Myrtle wanted to haunt the girl who laughed at her apperance. When the person dies, they go to limbo. It is a world inside their head where they get time to make the decision. Each persons version of limbo is completely different depending on the lives they led. But for people who have tampered with their souls such as Voldemort or Herpo the foul who split their souls into 7 and 1 horcrux(es). They will have to stay in limbo for eternity and live in pain for the suffering they have caused to their souls and the people they killed to make the horcruxes
Many fandoms say that they cant and some ghosts like Moaning Myrtle, Nearly headless nick and others never moved on. And it's stated that a soul chooses to return out of fear or unfinished business. I think that if a soul doesn't fear death but got a reasonable need to return to something then they may be granted the gift of moving on. Because Moaning Myrtle wanted to take revenge on a school bully and Nick was just scared of death. All crappy reasons. However, say if a soul chooses to come back to guide or protect a loved one or to aid someone then I think after the death of their loved one the ghost may be able to move on as their need will be completed. Like its just an example, and a theory.
I think there would be some 'condition' or extremely complex procedure that could probably allow ghosts to move 'on' into the other world. Given that even Voldemort himself had the opportunity to recover his soul from the stunted, mangled, divided soul it had become by undergoing intense remorse, then surely, SURELY, ghosts aren't surely beings who have done something so terrible that they don't deserve a second chance/a chance to move 'on'.
Well, perhaps if following Christianity, perhaps Christ could bless ghosts to move on. Or perhaps, an act of sacrifice or an act of extreme generosity could allow ghosts to move on. Given that wizards have succeeded in creating the elixir of immortality, like Nicholas Flamel, or horcruxes, or even the ressurection stone, various kinds magic that tamper with the fundamental laws of life and death, surely there would be some kind of magic that allow ghosts to move on, eventually.