Sympathctomies aren't a usual treatment for CRPS patients and aren't performed that often. If a patient with CRPS has their pain relieved by sympathetic nerve blocks, then specialists may consider a surgical sympathectomy, which is an operation to cut or remove some of the nerves.
This operation is very rarely used as there are specialists who believe the symptoms of CRPS may become worse in the patient.
According to Hooshmand, H. & Phillips, E. in their study article on CRPS and Sympathectomy,
"Sympathectomy may provide temporary pain relief, but after a few weeks to months it loses its effect. Sympathectomy and the application of Chemical Sympathectomy (neurolytic agents e.g. phenol, alcohol, etc.) should be limited to patients with life expectancies measured in weeks or months – e.g., cancer patients. Chemical Sympathectomy (e.g., alcohol, phenol or hypertonic saline nerve blocks) aimed at destroying the nerves are apt to fail, to cause serious complications, and aggravation of the pain – by leaving a large scar behind. Complex Regional Pain Syndrome (CRPS) patients should not be exposed to aggravation of pain due to sympathectomy, chemical sympathectomy or radiofrequency sympathectomy."
Within the same article, they conclude...
"Surgical procedures have no place in treatment of CRPS. Sympathectomy or removal of a part of the chain of sympathetic ganglia (on the side of the spine) has an extremely high rate of failure."