The lender with whom you're refinancing the auto loan has the rights to pull your credit report in order to check your credit situation. But the previous lender does not have any right to pull your credit report.
When you go to refinance even if it is with the same institution they will pull your credit. Your current lender will want to make sure you are still paying all your bills on time.Every time your credit is pulled it will knock down your score a few points not many.but your previous lender has no right.
Once you get a new loan, your credit report will show that your old loan has been paid off which will bring your score up more than the few points that you lost from your credit being pulled.