Every single thing in this universe affects something else. Every single pound of gas on Earth will affect the weather to some degree, even if it is .00000000000000000000000001 degree it still has an effect. To what degree is the question. If the amount is within acceptable tolerance that the Earth can counter-balance the effect, then we're good. But saying it doesn't affect the weather is probably not right.
If you ever want to prove a hypothesis, exaggerate one of the parameters. So if you shrunk the Earth to the size of a Pizza restaurant would all the pollution we emit in the world today affect the atmosphere then? Yes, without a doubt. So then make the Earth a little larger would it still? Yes. Keep making it larger. At what point would it STOP affecting it? Never. At what point would it be acceptable? Unknown.
My belief is that man made CO2 does not effect the weather at all.