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.