what happens if a spider bites a dog?

It depends on the kind of spider that it is, that bit the dog.

Depends on the spider. There are many venomous spiders that can potentially be fatal to people, they in turn can also be fatal to pets.

Brown recluse, black widow, among others.

Depends on what type of spider - if it's a dangerous one, it might, without treatment, kill.

If it's a brown recluse or black widow it is very dangerous. & They say the smallest spiders are also the most venomous.

This is an interesting question, and one that has been confusing me for a long time.

depends on which type the spider IS .

It depends on the species of spider spider

WHat kind of spider?

Dogs and other animals are much immune to bites like spiders. Like humans, some can have allergies to venomous bites and in rare cases, need treatment.
The hard part here is how you might know if a spider bit your pet, your pet can't tell you and you cant tell from any injury.
Another myth that needs debunking here is just because a spider can be medically significant to humans, doesn't mean that is the case with dog and cats. Their immune system is entirely different and they are not humans.
My own black lab was always nosing those big wolf spiders, and a couple times, lets out a little yelp, once even coming to me with the spider still on his nose. No injury, some very minor swelling in the snout and still didn't stop him from nosing the spiders.
Spider venom, is not geared for larger animals, they can kill another bug fairly fast, but has little or no effect on larger animals.
One example is in Australia one spider is 5 times more toxic to humans than the black widow, yet a common mouse there is literally immune to their bite.
My best guess is nothing at all will happen, and the spider is more at risk from your dog or cat.
Update:
Amongst all the misinformation, If 'anybody' can show me "PROOF" to the contrary, list your references.
Mine begin with UCRiverside, UC Davis, University of Washington, Dr. Philip Anderson, Dr Richard Vetter.