Foaming oil usually isn't a very good lubricant.  
There are several things that can be wrong.

Is the gear pump throttled in any way?
Does the pump have an internal bypass?
Is your gear pump sucking in air through the seal on the pump?
Is there a heater in the oil reservoir?
Does the oil quickly separate on standing?
What is the oil temperature in the gearbox and reserviour?
 

Is the gear pump throttled in any way? No. The oil does pass through a filter with relatively low pressure drop.

Does the pump have an internal bypass? I don't believe so.

Is your gear pump sucking in air through the seal on the pump? I will get the pump seal inspected.

Is there a heater in the oil reservoir? No.

Does the oil quickly separate on standing? No, slowly.

What is the oil temperature in the gearbox and reserviour? They are about the same.  The temperature starts to rise from ambient due to heat build up.  Eventually the machine shuts down on high oil temperature.

Right off check the desiccant to see its restricting air into the reservoir while running.  This should equalize after a few minutes, but check it any way.  Make sure the desiccant is still where it's supposed to be.

Check and see where the pump suction is located in the reservoir and is not restricted.  

I would look around for one of the small packaged oil coolers that are available.

Is the gear pump in good shape, not worn?

What is the viscosity of the oil?

This not a steam turbine driven gearbox by any chance?

I don't believe we have air leakage at the pump seal or a restriction from the dessicant.  There aren't any air bubbles in the oil when circulating with the gear box off.

The oil suction line isn't restricted and there is no vortex.

We are considering an oil cooler, but are reluctant add a potential water source in the form of a leak.

The gear pump was replaced when we repaired the gear box.

The oil viscosity is 220 cSt at 40 oC.

The oil pump is driven by an electric motor.

No bubbles until the gear box is operating, but the oil heats up prior to going into the box.

Is this after a few minutes running or is there a  temperature rise on cold start without the box running?

If on a cold start and I'm understanding you correctly you have to be putting the Heat(work)into the oil across the pump prior to the box.  

Look at pump specification sheet and check it against your motor speed. Look at the inlet and outlet pipe connections.
Also look at the pump specs to see if you have the right pump.  

Was anything changed in the gearbox during overhaul, like a splash shield, drip tube?

Have you changed the way the gears are physically lubricated?

Could you still be getting a little water into to the gearbox?

Get you lubrication supplier involved also. There are several things you might do with the lubrication oil.  Do you have the facilities to check and see if there is water, it may be emulsified, in the oil?  It can be done very simply if you have a lab.

Is this the same oil you used with an internal sump?

What caused the box to fail?

A lot of questions.

it sounds to me that you have a problem to get rid of entrained air in the oil. the air somehow gets in the oil, either in the pump stage (unlikely because you have no problem when the pump is running and the gearbox is not)or in the gearbox itself. A possibility might be too high an oil level there, with as a result gears churning through the oil. effectively acting as a mixer. Once you get air entrained in the oil, given it's viscosity and operating temperature, it may take quite some time to get rid of it.

I suggest you carefully check the oil level when operating the gearbox and also check exactly at which point the oil is introduced to the gears to be lubricated. Is it actually splash lubrication (oil picked up by the gears) or do you introduce the oil directly to the gears by directing a jet of oil on the parts to be lubricated?

One last thing to be checked: gear oils tend to loose their air-release properties when contaminated - and not only when water is present. Is there any contamination in the oil from outside?