Not all restaurants carry a liquor license, but some will give you the option to "bring your own bottle" when you sit down to dine. This is a courtesy on their part (when permitted by law) and not an obligation.
The same holds true for eating establishments that hold a liquor license. Yes, you can bring your own bottle (if allowed by law and the management), but the restaurant is not obligated to let you do so.
Why the reluctance?
For a licensed establishment, a good portion of the profit structure rests within the beverage alcohol category. If you bring your own bottle, they lose serious coin.
For this reason, most licensed restaurants will charge a significant "corkage fee" for allowing you the privilege to BYOB. It is not unheard of to see a $25 surcharge on the check when you bring your own wine to a licensed establishment.
On the other hand, an unlicensed establishment usually charges a nominal fee ($3-5). Their profit structure does not include beverage alcohol, so a smaller surcharge is enough to cover their service costs.
Wine service takes time, and time is money. When you BYOB, appropriate glassware must be brought to the table and an ice bucket if needed. Naturally, the wine will require opening. Sometimes it will merit decanting.
When a restaurant offers the BYOB option, it should be prepared to provide all of these services, especially if it possesses a liquor license already, and the wine service should be completely up to standard, i.e., no negligence just because you have brought your own.
Usually, the server will be quite attentive as most guests that bring their own wine to a licensed establishment bring something old or expensive.
And, it is completely appropriate for you to include the server (of legal drinking age) in the ritual of the "first sip." Ask him or her to fetch an extra glass so that he or she can taste too. This usually earns you big points and an evening of impeccable service.
It is considered very, very poor form to bring an item that already exists on the wine list. In fact, some restaurants will not allow this at all.
It is also considered very poor form to bring a bargain-basement bottle to an upscale establishment. This insults the chef and beverage director alike. Both go to great pains to put together a food menu and wine list that make both wine and food show to best advantage.
If you bring Two Buck Chuck through the door, you will pay top dollar for the privilege of drinking it and your server's smile will be skin deep at best.
You don't have these worries with a non-licensed account. Bring what you like. Nobody cares whether the bottle is rare and special or recently purchased in a close-out sale. The server will usually take no interest in the bottle he or she is opening and table service will be perfunctory. You might even have to pour for yourself or your guests after the bottle is opened. Expect a more casual atmosphere. Be pleasantly surprised if you get more by way of service.
In the case of casual BYOB eateries, it is completely appropriate to walk through the door with a paper-bag wrapped bottle. Many BYOB restaurants are located within short walking distance of a liquor store, and that is how most of the bottles walk through the door. For upscale restaurants, a wine tote would be more appropriate.
Always call ahead to know the fees, legalities and the policies of your BYOB restaurant so that there are no surprises. As "corkage" is already a service fee, you are not obligated to tip on this charge, but as with all things, good service should be rewarded appropriately. The corkage fee goes to the restaurant. Be sure to take care of your server.