If you want to retain information about a case but you do not want the case to count as a charge against the client's balance, you can zero out the invoice.
The only way to delete an invoice is to void the case and rollback the invoice.
1) Go to Billing > Invoices
2) Click the Gear to the right of the invoice and choose "Edit"
3) Zero out amounts and click "Update"
4) Invoice will now show as "Paid" from the date it was zeroed out