When you have a big load, when you need to accelerate quickly at speed (30+mph), that's about it.
2 ways it affects performance.
1. By locking up, it lowers your rpm (increases has mileage)
2. Takes longer to accelerate because the torque convertor has to unlock (to rev into the power band

)
The torque convertor basically works like a clutch, but grips more at higher rpm.
Scenario:
Imagine driving a manual tranny car.
you're doing 30mph in 5th gear.
You step on the gas.
Not much happens, right?
You shift into 4th, it picks up a bit better.
That's like the difference between Locked and unlocked
Another example
700rpm (idle) Very LITTLE grip.
900rpm SOME grip.
1200rpm More grip.
1500rpm MAXIMUM grip
As the rpm's go up, the grip stays the same.
Note: The torque convertor uses the tranny fluid to transfer the engine's power, so it can't be burned out....but the auto tranny has many other smaller clutch's/gears/actuators inside of it that can break (that's why doing neutral drops is BAD in an auto tranny car)