Thanks in advance,
A standard rub after a brine would likely make for a saltier product than I would like. You could make your own custom rub with little or no salt. On chickens/turkeys I keep things simple: melted butter with lemon juice, salt, pepper, a little heat (if I'm in the mood). You could just eliminate the salt on the brined birds. Keep us posted...
It would really depend on the amount of salt in the rub. If your using a fairly low salt content in your brine you should be fine. If the brine has a big amount of salt in it I would do as James and either make a rub with little to no salt or just season with salt, pepper, and some garlic maybe a little Rosemary
It seems like I always decide on the spur of the moment to smoke a chicken, so I haven't planned ahead enough to brine it.
So lately I've been just injecting with chicken broth, melted butter, and Montreal Chicken Seasoning, and skipping the brine.
Rub with some EVOO & dust with Montreal Chicken Seasoning.
Then smoke as beer can chicken with the injection liquid in the beer can.