No. At least all the FFL that I have talked with here in SC have said that. The only thing they are responsible for and charging you for is recording the transfer of the gun. With that said I did hear of one company that actually transferred the gun to the FFL and recorded it as a sale to the FFL.In that case the FFL was selling it to you and had to charge sales tax. In this case the FFL was responsible for collecting the money from you and paying the original seller. If you pay whoever you are buying it from directly, such as Bud's Gun Shop,then no sales taxes are charged.
Please note that in many states you are supposed to report that sale on your state income tax forms and pay the sales taxes on it when you file you state income taxes.