This ad from the 1974 Southern Bell Yellow Pages is similar to other ads for the Flaming Pit that I have seen in that it never gives a real street address for the place. That, and a complete lapse of mind that let me confuse I-20 with I-26 led me to originally post some pictures of what was certainly the wrong building for the place!
We never went to the Flaming Pit, though I seem to remember radio commercials that made it sound like a very exotic place. (I didn't do "exotic"..). I'm not exactly sure what "Open Hearth Charcoal Cookery" is, but combing that with the name of the place, I envision a large central grill in the middle of the dining area (hopefully very well ventilated with charcoal involved..) where your orders were cooked as you watched.
I'm a little surprised that the bar was The Wells Fargo Lounge given that Wells Fargo is, in addition to its evocative history, still around and presumably trademarked..
UPDATE 15 June 2010: Removed incorrect pictures, and corrected the text in which I was completely wrong about the location!