They're going to run into big problems related to overproduction and - assuming they try to address that - oil smuggling. As noted above, the United States will in all likelihood have the technical know-how and liquid capital necessary to really get production going, so you'll likely see some strong feelings on Yankees literally siphoning off the wealth of the Confederacy. Beyond that, a lot depends on when they find it. Spindletop OTL was in 1901, but I can imagine the Confederates finding it much later, which might not be to their detriment. Sort of a Dixie Arabia scenario.
It's hard to believe that the massive profits wouldn't awaken the avarice of the Confederal government. I think whenever oil is discovered, it's going to lead to great friction between Richmond and Austin.