"The premises" of a business or institution are all the buildings and land that it occupies on one site.* In your sentence, the book festival will probably be held somewhere ON the land around the school, so ON the premises would be correct.
"IN the premises" would accurately describe an event happening within the school building.
"On the premises" includes the area both within a building and on the outside. It is more general than "in the premises." It could refer to the area both outside and inside, or just outside, or just inside, whereas "in the premises" refers only to one inside area.
Google has 496,000 examples of "on the premises," like these:
"¢ ... cafeteria, lunchroom, lunch counter, soda fountain, or other facility principally engaged in selling food for consumption on the premises
, including, but not ...
"¢ ... Swimming Pools You don't have to take to the beach to enjoy a swim. The Rosarito Beach Hotel has two swimming pools on the premises
. ... www.rosaritohtl.com/otpremi.html
Google has 108,000 examples of "in the premises," like these:
"¢ ... the fire authority must be satisfied that the means of escape in case of fire, means the fire fighting and means of giving persons in the premises
warning in ... www.healthandsafety.co.uk/firep.htm
"¢ ... A customers' own alcoholic beverages which have been left in the premises
for storing have to be kept clearly separate from the licensed beverages in a ...www.sttv.fi/Alkoholi/alcoholissues.htm
There are 140,000 examples of "on the [adjective] premises," like these:
"¢ ... Original: English Third session The Hague 6-10 September 2004 Report to the Assembly of States Parties regarding discussions on the permanent premises
of the ... www.icc-cpi.int/library/statesparties/
... Insurer. The company (or most recent company, in the case of an Escrowed license)holding an insurance policy on the licensed premiseswww.michigan.gov/documents/
There are 46,000 examples of "in the [adjective] premises," like these:
"¢ ... the number of employees of NAUMEN Corporation due to course of action we took during last couple of years while we still were in the old premises
demanded from ... www.naumen.com/go/company/news/obj0172103140
"¢ ... A landlord is not ordinarily liable to a tenant or guest of a tenant for injuries from a hazardous condition in the leased premises
that comes into existence ... www.jocs-law.com/dutyof.htm
The examples above show "in the premises" to refer to a building alone. Your sentence would most likely be accurate with "on the premises."
*The Collins COBUILD English Dictionary. Harper Collins. 1995