This is one of the biggest no-nos due to safety and legal reasons. A police officer could stop and arrest you, no questions asked. Large oversize fursuit heads just don't allow you to see much, a critical item when driving. However, if you are wearing a facial appliance/prosthetic that doesn't affect much, if at all, of your vision, and you are able to see very well, you may be able to do so. However, you will have to be aware of other drivers, who will not be expecting this, and the results could be anything from quite hilarious to not funny at all.
If you have doubts about your ability to drive, have someone drive you. The fursuiter can quite easily wave to the other drivers and younger kids if they're out there.
The same goes for using motorcyles and bicycles and other related items. Do not drive them unless you're in a safe environment.
If you merely have straight makeup, and small appliances like Woochie or Cinema Secrets, you should be ok. boogi can easily drive while in Mutant Cat, due to the way the facial appliance is created. There's no large muzzle to deal with, and the vision is unrestricted.
In short, talk with your local police, describing what you want to do, down to showing them what you will be wearing if necessary. Most likely, they will allow straight makeup, and some facial appliances, since there are or will be no visual or audio hindrances.
It's no funny moment when you're arrested, and jailed, then have to appear before a judge in full makeup and appliances the next morning, if you never were released from jail. There's the story of one guy who was in full Klingon gear, right down to the headpieces and uniform. He was arrested for drunk driving, and was taken to jail. He had to explain to the jailers out there that he was unable to take off the makeup since the remover was in his car, now impounded. That plus other officers were giving the arresting officer some trouble, humming the Star Trek theme, making alien jokes, etc. They proceeded to take his mug shot, with a number of other officers having their pic taken with him. This guy is probably the only person out there who has a mug shot of him as a Klingon.