How does a catalytic converter affect a car's gas mileage?

First, no it's not legal. If it passed inspection, it's only because an Accent is has a very small, fuel efficient engine and it didn't emit enough to trigger a problem. Also, I'm guessing he took off the "downstream" catalytic converter. The "upstream" one is immediately off the exhaust manifold and is the "main" cat. The downstream cat is only there to catch stuff that the main one misses, which might not be much. It's illegal in all 50 states to modify the catalytic converter too much, which would include removing it.

As for the fuel mileage, no it would not improve it. If he took the main cat off too, he is probably harming fuel economy because the engine is not receiving the correct information regarding the fuel/air mixture, which is required for the CPU to decide how much fuel to add. There is a sensor after the main converter that feeds information back into the CPU to change the mixture to more optimal settings. It's possible that he reprogrammed the CPU to address this, but it still won't be operating nearly as efficiently as before. Keep in mind that a team of engineers designed that car to get as much fuel economy as possible (for the cost). If it was that easy to decrease gas usage, they would have found a way around it.

This ignores the harm to the environment. The cat removes most of the harmful gases that come from burning fuel.

Dumb move and would cause the engine light to come on. Gas mileage would get worse because the computer is not programed to run the engine at maximum efficiency with a code stored and a missing cat. It's not legal and it wouldn't have a hope of passing an emission test.