Several possible problems come to mind.
Ensure that the lamps at the main station and contestant stations are identical, this is important for the corrct operation of this circuit.
Are all the zener diodes the correct way round? Particularly the 15v zener.
Is your power supply 24v? If it is a bit low the circuit should work ok, but if it is high (26 or 27v) then you will get problems.
Does the problem only occur when the buzzer is sounding?
I can see a potential problem during the time the buzzer relay is energised, for a short time there will be enough voltage at the contestant stations for them to latch on when their button is pressed.
JimB