I've always wondered why folks that got brick driveways really liked them. I think you answered that for me - They look nice! (I'll get to your question in a moment ...)
And that's about where it ends. If you think about it, driveways are what we park our cars on. Our cars drip oil and all kids of fluids onto the driveway. At the same time, we shovel them, we chop on them, we essentially abuse them. GOOD THING we don't sleep on them, eat on them, or entertain our guests on them.
With this in mind, you can buy brick cleaners at your local Home Center. I don't think there are any Guarantees out there that say the touted-product will remove the 'stain' in its entirety. You see, brick is a pourous material. It will ABSORBE all that junk your car, or whatever, drips/spills on your driveway. The best you can do is TRY to keep up w/it. Visit the home centers for cleaners and consider power-washing the driveway once a year. I'd suspect that before long you'll be asking how to get rid of all those weeds and all that grass that's growing up between the bricks. Good luck! I wish I had a better answer for you.