Sure, for the same reason that all relationship based selling systems work. I will qualify my answer by suggesting that the individuals within the dealership should act as individuals, not inside of a dealer sponsored page.
People like to do business with people tnat they like and facebook allows you to make friends before you try to sell them a car or service their present one. Old school "car guys" learned to deal with their spheres of influence which were limited to family and friends gathered from High School, College, The local Night Club, The Chamber of Commerce, The Boy Scouts - and the list goes on.
Faceook, like all the social networking sites that have developed on the WWW, is simply a virtual version of all of the above. Just like in the real world, if your conversations always start by trying to sell your friends a car you will lose your friends and the sale. Also, just like in the real world, if you focus on caring about your friends with only a casual mention that you happen to be in the car business then it is only a matter of time before they need a car at which time they will approach you to help them buy one. With that invitation, you are doing them the favor by selling them a car which coincidentally is in full view of your other friends on facebook who happen to be following your kind gesture --- which they will keep in mind when they need a car and the sales go on!!
After all, what are friends for!