Will I have any problems with a building inspector if I pull a permit to replace the pushmatic breaker panel but leave all the clothe wrapped wire (Assuming the cloth wrapped wire was to code when it was installed)

Yep, you will run into a brick wall. The inspector will require you to bring wiring up to current code. Stuff is grandftherd if only minor changes or improvements are made, replacing the panel is not minor.

I doubt the inspector will require a home to be re-wired just because a service was upgraded.

I have done many, and never had to upgrade the wiring.

Ask the inspector before doing any work.

Yep, you will run into a brick wall. The inspector will require you to bring wiring up to current code. Stuff is grandftherd if only minor changes or improvements are made, replacing the panel is not minor.Sorry, I have to completely disagree here.

I have done probably well over 200 service changes in my career and not once id I have to upgrade the wiring in the house because of it.
The branch circuits are not being modified or changed because of a panel change.
Same thing goes for arc-faults. You do not have to install arc-fault breakers when doing a service upgrade or panel change in an older house.

And I have to also say, replacing a panel is pretty minor in the scope of things.

I doubt the inspector will require a home to be re-wired just because a service was upgraded.

I have done many, and never had to upgrade the wiring.

Ask the inspector before doing any work.
My answer is slanted by usaveu many posts yesterday. He is flipping houses with major remodeling.