If the induction motor is lightly loaded it will continue to run. As it nears its rated power, the current will rise because the power will be 2 x Iph x Vph x Cos Theta instead of 3. with V constant, I must rise. Add to this you mess up all the magnetic circuit and the efficiency goes for a ball of chalk. The induction motor will pulse as it goes through the rotation where it is not being driven. This may be small but it is there and will affect gearboxes and vee belts. If the motor is star connected the phasors move out and the neutral moves away from the star point. The voltage on each turn will drop and the torque will be much less. I expect the induction motor to stall if at full load and eventually burn out at slightly less than light load.
Your protection should detect "Single Phase" and stop the motor.
If your motor is connected to a variable frequency drive (VFD) and the input is removed, as long as it is not the phase of the VFD gets its power from, it will continue to run as normal. Some VFDs have a separate power supply connection and these will run no problem no matter which phase is missing.