You need a license for any piece of gear with an (L) next to its availability code. Beyond that, it's between you and your GM, and every GM is different.
Personally, I wouldn't require a separate driver's license. If you buy a vehicle, then I assume a relevant operator's license is part of your Fake SIN and Lifestyle expenses. For that matter, I don't ask players to buy a license for each individual item of restricted gear, either, just one per category (e.g., Firearms, Augmentations, Magic, Matrix gear). I'm a nice GM, and opposed to overly detailed bookkeeping on general principle.
I've had players ask if they could buy a fake license to give their character appropriately convincing credentials as a doctor, lawyer, and private investigator -- in every case I was happy to allow it, but it was their own idea. My only stipulations are that a fake license must be attached to a particular fake SIN (you can't just swap them around among different identities), and must be the same rating as that SIN (the rules on p.273 allow a fake license to have a lower rating than its SIN, but for simplicity we always make them the same).