Leweston School, near Sherborne, Dorset, announced it would not reopen after half term.
Leweston School gave no warning of Tuesday's shock announcement, Edward Morello MP says.
Police say six youths wearing balaclavas were caught trespassing on the grounds of Leweston School.
Leweston School boasts alumni including actress Kristin Scott Thomas and Baroness Dido Harding.
Sherborne school is sharing the father of modern computing's story with future generations.
Dorset Police says the fire did not spread and no injuries were sustained.
Chairman Nigel Chambers says the number of visitors had been declining steadily since Covid.
Erin Bloomfield plays and coaches at Sherborne FC, but also represents England on the sand.
The 'most significant upgrades' in over a century will be built within the Grade I medieval abbey.
