The answer to this question depends on the date you created your Stripe account.
For Stripe accounts created before Sept. 14th 2017
Nope! When you issue a refund against any payment collected by Stripe, Stripe will include their payment processing fee a part of the refund.
For Stripe accounts created after Sept. 14th 2017
Yes - Stripe changed their policy for newer accounts and will charge their processing fee as part of the refund.