How to increase the motor speed?

There are two things to consider when looking at increasing the shaft speed of a motor (by whatever method).

First: mechanical. Can integrity of rotor design be maintained? (Typical max safe speed is NP * 1.25 which does NOT mean it operate at that speed, but only that it will not come apart and cause injury.) Will bearing and/or lubrication handle the increased speed? (There's going to be more heat.) How about vibration? (Typically, faster speeds require smoother operation to prevent damage.)

Second: electrical. How does load torque increase with speed? (More torque means more current - which means more heat and thus shorter lifespan for windings and insulation.) Will motor actually be able to develop sufficient torque? (Turning faster means cutting more lines of flux, which increases magnetic saturation, which makes it harder to excite.)

Some motor can only withstand 50-60 Hz but, some can do upto hundred Hz frequencies. High frequency motors are specially designed with advanced materials and winding design. 50-60 Hz motor can get damaged if we increase to high frequency. Increasing frequency can also heat the machine up.

We have to discuss the specific application with the vendor of the motor that we're trying to operate at higher speed.

They will want to know answers to the following questions (as a minimum):
What is the motor driving?
What is the torque profile of the load? (Include everything from zero to desired speed)
What type of variable frequency drive (and whose) is being used?
What is the intended control scheme (constant Volt/Hz, constant current, etc.)?
What frequency and voltage output are you expecting from the VFD (e.g. input to motor) - at specific load points?
Are you intending to change the ventilation scheme (e.g. force cool)?
What are the "normal" minimum and maximum operating speeds going to be?
WHY DO YOU WANT TO MAKE THE CHANGE?

Without these answers, and a bit of time to investigate their design, I'm pretty sure the answer is going to be either: A) NO OVERSPEED CAPABILITY or B) YOU OWN IT, DO WHAT YOU WANT - BUT THE WARRANTY IS VOID.