Under what circumstances does a flying club need to do 100-hour inspections?
AOPA addresses this very matter in their question of the month found here


  A flying club aircraft normally would not need to receive 100-hour
  inspections.  FAR 91.409b requires a 100-hour inspection when a person
  receives instruction from a flight instructor who also provides the
  aircraft.  The FAA guidance further clarifies that the 100-hour is
  required when a person is providing flight instruction for hire in the
  instructor’s personal aircraft.


Your issues arise with these two points 


  
  The students (Receiving dual instruction) may or may not be members themeselves.
  A club rule says that only approved club flight instructors may offer instruction in club aircraft.
  


As the answer goes on to state 


  Some clubs require a flight instructor to be a member prior to giving
  instruction in club aircraft.  In this case, the FAA doesn’t view this
  as an issue as long as three conditions are met:  
  
  
  There must be no contractual relationship between the member-instructor and the flying club for the provision of flight
  instruction.
  The member-instructor is not recommended or given preference by the flying club. 
  The club members are free to choose instructors who are not club members. 
  
  
  If any one of these conditions is not met, then a 100-hour inspection
  would be required for the club aircraft.