I charged VAT on a product that should have no VAT. The invoice has been paid. I need to refund them the VAT. Can I just amend the VAT rate on the paid invoice?

Hello Invoices-t4sltd-, 

Welcome to the Community page, 

So what you would need to do is change the VAT rate, that will result in an over payment, you can then refund the money and record it as a customer refund for over payment in QuickBooks.  If you have a look at this link it shows you the steps on how to record a refund for your customer.