So I'm wondering on weapons with the burn quality, namely the flame projector, if extra successes on the initial hit will carry over to the burn damage or is just strictly the base damage is done as the burn?

Only the base damage as described in the weapon quality burn.

Only base damage for the Burn but the extra successes would add to the Blast effect on a successful attack.

oh cool thanks for the replies.

Not at all and good day to you sir.