Can white vinegar help the pain of first degree burns?

In that you're pouring a cool liquid over the burn, there is some relief. However, vinegar is acidic, and is not what I'd choose to treat a 1st degree burn. Cool, damp (with clean water) compresses -- towels even -- laid over the burned area can offer some relief. If the skin surface is not compromised, you might consider an over-the-counter benzocaine or lidocaine ointment. Note that this advice works only for 1st degree burns and not for more severe injuries.