Could Professor Binns be fired?
Technically, yes. The headmaster of Hogwarts can fire a professor and so can the Minister of Magic by introducing a reform in education by issuing an Educational Decree which authorizes a person appointed by the Minister to issue an Order of Dismissal. The Educational Decrees of 1995–1996 school year were cancelled, but nothing prevents the MoM to write new ones.
Binns doesn't seem to be a malevolent ghost out of control. He can be reasoned with and persuaded. For example initially he refuses to tell the class about the tale of the Chamber of Secrets, but hearing out Hermione, he relents:
“Please, sir, don’t legends always have a basis in fact?” Professor
Binns was looking at her in such amazement, Harry was sure no student
had ever interrupted him before, alive or dead. “Well,” said Professor
Binns slowly, “yes, one could argue that, I suppose.” He peered at
Hermione as though he had never seen a student properly before. ~Harry Potter and the Chamber of Secrets
So if fired, most likely there'll be no need to call the ghostbusters (old or revamped) to exorcise him. But even if for some reason Binns refuses to comply, we know from the accepted answers to these questions(1,2) that ghosts can be controlled and made to leave a particular location (for example the history of magic classroom) if needed.
Is there any hope that Professor Binns would be fired?
Very likely that there would be a drastic revision of the O.W.L exam requirements for History of Magic by the MoM after the Second Wizarding War.
After all, according to JKR there was revitalisation and reform in the Ministry which also included Hermione Granger advancing the rights of non-humans, and eradicating pro-pure-blood laws:
JKR: So Harry and Ron lead the way in recreating the new Auror Department.
I would imagine that Harry is heading up that department, which is not corrupt in any way. It's-- it's a really good place to be. And Hermione ... I think she's now pretty high up in the Department for Magical Law Enforcement. Where I
would imagine that her brainpower and-- and her knowledge of how the
dark arts operate would really give her a-- you know, a sound
grounding. So they're all at the ministry but this is a very new ministry. They made a new world.~Harry Potter: The Final Chapter" Dateline (NBC) , 29 July 2007.
and in another interview:
JKR: Hermione began her post-Hogwarts career at the Department for the
Regulation and Control of Magical Creatures where she was instrumental
in greatly improving life for house-elves and their ilk. She then
moved to the Department of Magical Law Enforcement where she was a
progressive voice who ensured the eradication of oppressive,
pro-pureblood laws." ~ Webchat with J. K. Rowling, 30 July 2007 www.bloomsbury.com
So a new world order it is then, which usually comes with reform in education, especially in the discipline of history. New textbooks are written, ones that revise and challenge the orthodox views and the outdated social hierarchy, present different sources and interpretations to historical events and social issues previously omitted. In the HPU the new books would also include the causes of the Second Wizarding War, the events that occurred and so on.
We know Hermione's views about biased or outdated history books:
"It's all in Hogwarts: A History. Though of course, that book's not
entirely reliable. "A Revised History of Hogwarts" would be a more
accurate title. Or "A Highly Biased and Selective History of Hogwarts,
Which Glosses Over the Nastier Aspects of the School". ~Harry Potter
and the Goblet of Fire
and now that she is
the Minister of Magic
If it isn't done already, now nothing could stop her if she wishes to do an educational reform.
This would surely require from the current headmistress of Hogwarts, Professor McGonagall to revise the old fashioned course of History of Magic, which will include replacing the old books written by Bathilda Bagshot and most likely also hire a new history of magic teacher.
And if he were fired, what would happen to him? I hope he'd be ok...
If fired, I think Binns probably'd be offended. As far as he is concerned, he is teaching history according to his best ability and is not aware how much room there for improvement:
“That will do,” he said sharply. “It is a myth! It does not exist!
There is not a shred of evidence that Slytherin ever built so much as
a secret broom cupboard! I regret telling you such a foolish story! We
will return, if you please, to history, to solid, believable,
verifiable fact!” And within five minutes, the class had sunk back
into its usual torpor.~Harry Potter and the Chamber of Secrets
We know ghosts can be offended or disappointed by the treatment they got after their death: Nick was hurt when his application to join the Headless Hunt was denied by Sir Patrick and Myrtle was offended...um, most of the time.
The current headmistress isn't a cruel person either, so she'd probably find a diplomatic solution. Maybe the new professor and Binns would split up the classes with Binns teaching the first- and second years for example, or the new teacher will be his 'assistant'.
Or indeed he'd be dismissed in a tactful manner, as someone who deserves a rest after who knows how many decades or centuries of teaching and offered to stay at Hogwarts in his current living quarters.
but I'd be sorry to see Binns go. JKR's descriptions of his lessons were hilarious and reminded me of my professor of statistic, and how we suffered during his lectures. Teachers are not perfect, many of us had people like Snape, Binns, or Lockhart teaching us, that's why these characters are so amusing. And firing the only ghost who has a job seems such a pity...