Success rate for liver shunt surgery?

I am assuming you're talking about surgical correction of portosystemic shunts in dogs.

This depends on the nature of the shunt. Some liver shunts are multiple very small vessels ("microvascular") and surgery is not a good option. Some are easily found outside of the liver ("extra-hepatic") and surgery can be a great easy fix, others are "intra- hepatic" and surgery can be more challenging but still the best option.

Proper diagnosis of the nature of the liver shunt and patient blood values will help determine your prognosis and success rate. If your dog already is very sick they can have somewhat rocky recoveries vs dogs that are feeling well maintained medically on low protein diets etc..