I have a customer who has a fixed displacement gear pump that is bypassing about 2-3 gpm under load. I recommended replacing the pump but their solution was to increase rpm to get to the required flow under load. My concern is the heat generation and eventual failure of the pump.

Will the pump wear faster now the speed is increase? What is the failure mode for a gear pump?

This begs a couple of questions. What is the pump displacement?  Without this, we do not know what the percentage loss is.
A 60 gpm "efficient" piston pump will have have 3 gpm (5%) heading out the case drain when it is new.
If the pump is worn, then increasing the rpm is an expensive short term fix. The additional lost flow at pressure will take more fuel of kW's and damage the fluid.  Existing contamination will cause wear and the need to increase the rpm even more.

Have you tried using a peristaltic pump??

What is the specs of the pump motor assembly?
if it's a low flow pump and its dumping oil back to suction under load, i don't think increase in RPM will help as the load will remain same.
replacing a pump/repair the existing one would be a better option.

Hi Kamlesh -  Dan Helgerson answered a similar question a while back, he stated “This begs a couple of questions. What is the pump displacement?  Without this, we do not know what the percentage loss is. A 60 gpm "efficient" piston pump will have have 3 gpm (5%) heading out the case drain when it is new. If the pump is worn, then increasing the rpm is an expensive short term fix. The additional lost flow at pressure will take more fuel of kW's and damage the fluid.  Existing contamination will cause wear and the need to increase the rpm even more”.  

Another suggestion was to use a peristaltic pump.  Those are generally used when you don't want a possibility of contaminants in the fluid and they are commonly used in the medical industry as an accurate metering pump but they are limited as to their maximum pressure that they can supply flow at.

I would advise that to answer your questions you need to understand more about the physics of the failure mode.  A short term solution can be to run the pump faster to make up the flow, but as you noted you might be accelerating a wear mechanism that is leading to the leakage or generating problematic additional heat.  Is it possible to get a look inside the pump?