Does she say “put it AT B tier”? Thanks.

No, they say "put it B tier"

I think it's "putting it B tier", but she's not enunciating all that clearly so I'm not sure.

She said "putting it B tier." While not proper grammar, this is how people speak, shortening the full sentence. "I am putting it in the B tier" naturally shortens to "putting it B tier." If she had said "putting it at b tier" , that would mean the same thing, and is similarly a non-grammatical shortening that is commonly done and understood.