Did you manage this? You need a MAS extractor utility.
Firstly...BACKUP YOUR ORIGINAL FILE!
When you open up the correct MAS file (ARC_CAMARO_MAPS.MAS), you'll see three tyre DDS files in the list.
ARC_CAM_TYRES_000.dds
ARC_CAM_TYRES_001.dds
ARC_CAM_TYRES_002.dds
Extract those three files onto the desktop.
Now this is where it gets a bit tricky....
Each DDS file is the tyre at various speeds of rotation. So "000" is static, "001" is starting to roll and "002" is at speed.
Using Photoshop (or maybe GIMP if it has a radial blur option) you need to use the static version ("000") and apply your Goodyear logo using the bend tool in PS. Thankfully the tyre is already blank which is a great bonus.
Then duplicate the whole set of layers (including the background), apply radial and motion blurs to each one so you have various levels of movement.
Save each one back out as the three DDS files. Go back into you MAS extractor utility, delete the three original files and replace them with the three DDS files you've created.
It isn't a quick job I'm afraid, I did Yokohama on mine and it probably took about an hour or so, playing with the layout, making sure it looked right, etc. When lining up make sure the logo is about 35% from the outside edge to start with. I put mine in the middle and it was touching the wheel rim.
But I must stress you backup the file first. The MAS utility can go wrong and will damage the file. It did it to me three times and simply causes the game to crash when it tried loading the cars.
The other tip I can give is just concentrate on getting the static one lined up first, then move onto the motion blur.
You can just do the static one by itself if you like but as soon as the car moves the logos will disappear. Which can look a bit weird.