First of all, earth doesn't have the same atmosphere it had, back when dinosaurs walked the earth, when most of current carbon-deposits formed. Some atmospheres earth had would even be toxic to humans. So, you're right, changing the atmosphere at the speed we're doing it is the problem, not the change in general.
Second, yes, earth is a rather closed system (with minor leaks, but anyway). Still, the timescale is important.
Let's go with a simple allegory:
Say, your friend had a pickup-truck (sorry, in my mind deniers all drive SUV or pickup^^). He uses it to transport a lot of material, e.g. paving stones. He knows, his truck is able to transport the total amount of stones (because he tried in the past, or because the specifications say so, or because someone told him, doesn't matter - like we know that earth's atmosphere held a lot more CO2 back in the day).
Is he going to just dump the whole load at once from 2 m height into the back of his truck? I doubt so, because that would at least put some dents in the truck, or might even break the springs or an axle.
He'd pack the stones there manually and use enough padding to make sure nothing gets damaged or slips during the transport.
Currently we're dumping the stones all at once - and we'll surely break the vehicle, if we really continue. Earth, like the truck, needs to adapt to changes. And, as we know, geological timescales are a lot longer than we can truly observe - thus the current changes, though stretching over several generations, are way too fast for earth to really adapt.
Third: While earth is a closed system, states within a closed system may change. Say, you have a terrarium with some mice and a lot of wood. Then you burn the wood. The mice might survive the fire, but in the end will suffocate, since there is a lot more CO2 in their atmosphere now.
So, yes, releasing trapped carbon from fossile fuels and whatnot to the atmosphere doesn't change the overall mass of system earth. But it changes the state of the system - and we are depending on the current state of the system.
Also - I know, this is not scientifically - say we are wrong when it comes to greenhouse gasses, and he is right. What if we undertook a lot of effort to not to produce them, to clean our environment, to create less waste? We'd wake up one morning and be "Dang, we made our planet worth living on again, how could we?!" - Or, as one of my professors put it: "Would you get on a plane when there is a 10 % chance, the plane is going to crash?"