How is ATP production affected by a inner mitochondrial membrane that is leaky to protons?

Yes.

Uncoupling agents -
 
These make the inner membrane permeable to protons. The proton gradient leaks away bypassing the ATP synthase (electron tramsport has been uncoupled from ATP synthesis). The energy released is wasted as heat instead of being stored as ATP.

That is correct.

Remember how ATP is formed...

The electron transport chain causes protons to be "pumped" out of the matrix and into the intermembrane space.

Normally, this causes a PROTON GRADIENT.

To reenter the matrix and restore equilibrium, the protons must pass through ATP Synthase. The energy created upon reentry is used to synthesize ATP from ADP and Pi.

But... what if the membrane has a "leak?"

Then it's simple - the proton gradient can not be generated because the protons are able to maintain equilibrium by passing freely through the leaky membrane.

This means that they are not utilizing ATP Synthase for reentry, therefore ATP can not be synthesized.