I've been binding only occasionally. When I do, I'm just using a smallish sports bra and a smallish athletic shirt. It's not so tight that my breathing is constricted or even uncomfortable.
But afterwards it's crazy itchy all over the boobs.
Could this be a binding thing? What to do?

It is usually just sweat and the material.

I agree that it might be the material. I like to wear an under armour shirt under my binder and when I do I get a little itchy if I sweat more. It happens less when I wear just a normal cotton shirt underneath it.

I always wore a tank top under mine and it greatly reduced the itching.

My back itches a lot. I wondered if it was something to do with blood being able to flow there easier, but sweat makes sense too.

I don't seem to be sweaty everywhere I itch. Weird.

Usually itch in the middle of my back, at work, where I don't have my back scratcher. Then I get to play Baloo and rub up against some wall and hope no one notices.