No, just take $A$ and $B$ to be intervals, then it's clearly not true that $\mu(A \cap T^n B) - \mu(A) \mu(B)$  is small on average since $T^n$ is uniformly distributed among rotations of the unit circle (for all $n>0$) and $\mu(A \cap T^n B) - \mu(A) \mu(B)$ is large for some positive fraction of rotations.