It's an intriguing idea, but if it worked, I think people would've figured this particular question out a long time ago.

Actually, assuming that there's any similarity, brining a turkey makes it more moist and juicy as some of the salt penetrates the bird. Drawing more fluid into the affected area seems to be the opposite of what you think it is going to do. The salt in the brine also serves to help break down the turkey meat. If you've been bitten with venom that is trying to break down your tissues, helping it along also doesn't seem to be a wise idea.

Besides, even if you were able to somehow only draw out fluid from the swollen bite site and that's all, I'm still not sure that's the best thing to do. Any remaining venom and toxins that your damaged cells are releasing as they die and split open would tend to get more concentrated as the water content in the swollen area decreases. It doesn't matter after some point, but again, I don't think it helps any.

But that's just my Food Network-based opinion, so I'm all ears on alternate answers.