Mark said, "it would ruin our school's reputation!"
If I want to quote Mark here,
Mark noted that doing so and so would "ruin _ school's reputation."
If I want to emphasize that Mark identifies himself as part of the school by using "our," can I say
Mark noted that doing so and so would "ruin our school's reputation."
Is it wrong since "our" doesn't agree with "Mark"? I don't think it makes sense to change it to "his" as the school is not "his."
You have the following choices. Which one to use depends on the context of the rest of what you are writing and what your readers can be assumed to know.
Mark noted that doing so and so would "ruin our school's reputation."
...a direct quote, and definitely the best and safest choice - though it may leave some confusion if reader's don't have a context for who "our" refers to.
Mark noted that doing so and so would "ruin [his] school's reputation."
Mark noted that doing so and so would "ruin [their] school's reputation."
Mark noted that doing so and so would "ruin [the] school's reputation."
...could alternatively be used but have the disadvantage of slightly modifying the original quote. But you might choose one of these if your readers don't know the context of whose school is being referred to.
Note that if you modify the word "our", be sure to enclose the new word in [brackets] so readers know that you have changed Mark's exact words.
Mark noted that doing so and so would "ruin our school's reputation."
Is it wrong since "our" doesn't agree with "Mark"?
I don't agree that "Mark" and "our" don't agree. When Mark refers to "our school," that means the school that Mark belongs to – along with everyone else affiliated with the school. That's why Mark said "our" and not "my"; he wanted to emphasize collective ownership.
The assertion that "Mark" and "our" don't agree implies that one person should never refer to "our anything" – not unless at least two people are speaking in unison. But that's not the case:
Said the captain: "Sailing too fast through these ice floes could sink our ship."
Said the football coach: "Too many red cards caused our team to lose."
Said Mark: "Hiring that professor would ruin our school's reputation."
Now, the reporters go to work:
The captain said that reckless sailing through the ice floes "could sink our ship."
The coach said too many penalties "caused our team to lose."
Mark said that hiring Professor Clemons would "ruin our school's reputation."
I don't see any grammatical problems here.
That said, you might consider changing the "ours" if you omitted the quotes altogether:
The captain warned that reckless sailing through the ice might sink their ship.
The coach mentioned that too many penalties caused his team to lose.
Mark argued that hiring Professor Clemons would ruin the school's reputation.
Bizarrely, I'd say it depends whether you're writing fiction or non-fiction.
Using [the] or [his] in fiction is just going to look strange, so just ignore them (or better yet, rewrite Mark's original quote to make it easier) - but in non-fiction they're more factually accurate to his quote.
All of that doesn't even get into whether the narrator would identify as a member of that school himself...
