If you have not read the Half-Blood Prince, please stop now!
This analysis contains spoilers!
Lots of them actually.
Will Hogwarts remain open?
“Very well,” said Professor McGonagall, glancing out the window at the grounds, checking to see whether the Minister was yet approaching, “then I must agree with Filius that the right thing to do is consult the governors, who will make the final decision.” (Half-Blood Prince, p629)
At the end of Half-Blood Prince, the future of Hogwarts lies in question. What will the governors say to McGonagall’s request? It is very clear over the course of six books that the Board of Governors is relatively fond of Dumbledore, and as divided as the staff appears to be on the issue, they will likely want keep the school open, dedicated to teaching as few or as many that wish to return.
It is very unlikely Hogwarts’ board of governors will choose to close the school. Despite numerous scandals and opportunities over the years the governors have had an opportunity to shut the school down, they have not:
The Sorcerer’s Stone. Voldemort taking over a teacher! The school placed in jeopardy by hiding the Philosopher’s Stone. Does it get any more perilous than that? (Sadly, yes it has.) The board of governors did nothing; the Ministry did nothing.
The Chamber of Secrets. Students were being attacked, and the best they managed was suspending Dumbledore for a time (they were coerced by Malfoy, however). Fudge actually stuck up for Dumbledore at this time. (p262)
The Goblet of Fire. A student died. The Ministry began to turn against Dumbledore, largely because they were embarrassed to be proven incorrect over Voldemort, not out of any concern for the student population at large. The board of governors did nothing, though.
The Order of the Phoenix. Again, Dumbledore was suspended for a time by the Ministry of Magic, not the governors.
As cruel as it may sound, the Ministry is no fan of Dumbledore (Scrimgour proves this), and the school may remain open on the will of the Ministry but against the school board governors wishes. The governors are likely to want to go with the wishes of parents in this matter. They are presumably parents and former students of the school. The Board of Governors may choose at this time to make a show of solidarity against Voldemort, and it has been stated on a couple occasions that Dumbledore will never truly be gone from the school so long as there are people still loyal to him there. Barring any major disaster wiping out the entire staff, this will always be the case. Additonally, Hogwarts remaining open will most likely have the following effects upon Harry and the plot of the remaining book:
Harry will have some place to reference when he needs information he cannot get ‘out on the road’ hunting Horcruxes.
The houses will have to combine – or the houses simply will not matter, as there will not be enough students to ‘compete’ against one another. Seamus was mentioned to be fighting his mother to stay – there are certainly other students in that position. However, since they are not all adults, they may not have a choice. Some sixth and seventh years may turn seventeen and as such, be able to return on their own, however.
Quidditch will become irrelevant (sorry, sports fans! The presence of Quidditch by and large has only been used to effectively demonstrate Harry’s natural knack for flying. The rest has been just bonus and superficial character development.)
Hogwarts will probably not accept new students.
More attacks at Hogwarts are inevitable, but the school will not be destroyed. Many of the staff will probably die. It is very likely those who remain behind (students, one or two staff members) will take up the mantle to “rebuild Hogwarts” when Voldemort is finally defeated.
Hogwarts remaining open also provides a place for Luna, Ginny and Neville to play out their parts. Just because Harry is taking the show on the road doesn’t mean that each of them won’t have a part to manage in all of things or that they won’t appear in “Harry’s Grand Adventure” (as I’ve come to think of it). Closing the school would to pare down the cast to the essentials, but massive deaths can also accomplish the same thing. Killing off a lot of those people would be depressing and necessary, but that has never stopped JK Rowling before.
Hermione will try to combine school and helping Harry, but eventually abandon Hogwarts. Her friendship for Harry and loyalty as such almost demands it. (Also, if Ron goes – and he will, that’s just extra incentive.)
Hogwarts has a role to play in the Horcruxes due to Hufflepuff’s cup and Slytherin’s necklace.
It is likely another Horcrux is located at Hogwarts.
Once things “calm down” and the initial shock wears off, the future of Hogwarts really isn’t in question. It is actually most likely the staff will choose to keep Hogwarts open, despite pressure from worried parents, government bodies, and the like to possibly do otherwise. By the time we see everyone again, it will probably have become a non-issue, as most of the remaining book will not need to take place at Hogwarts to finish out the story.