Most organizations routinely occupy the premises of others for varied purposes.  Such 'tenancy' can lead to financial responsibility to the owner of the premise should negligence result in damage to the facility or its contents.

Hotel Room

The employee of an association while travelling to a convention fell asleep while smoking in a hotel room.  Damage was luckily contained to the room by the hotel's sprinkler system, however the repair was still quite substantial.

Rented Hall

A youth club was held responsible when the town hall used for teen dances was maliciously vandalized by 'a few bad apples' during an event.  The organization had attempted to defend itself on the basis that the chaperones provided reasonable supervision, however the court disagreed. 

Kitchen Fire

While preparing meals for a social gathering later in the day, a volunteer caused a grease fire.  The fire extinguishing system was not in proper working order, thus resulting in serious damage to the facility.  The organization attempted to deny liability on the basis of the faulty extinguisher but was found mostly to blame for allowing the volunteer to use the deep fryer without appropriate instruction.

Some examples may be generic in nature, applicable to all classes of business - not specifically non-profit organizations.