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.

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