The Karp–Lipton Theoem states that if $\mathsf{NP} \subset \mathsf{P/poly}$, then $\mathsf{PH}$ collapses to $\mathsf{\Sigma^P_2}$. Therefore, assuming separations between $\mathsf{\Sigma^P_2}$ and $\mathsf{\Sigma^P_3}$, no $\mathsf{NP}$-complete problem will belong to $\mathsf{P/poly}$.
I'm interested in the following question:
Assuming that $\mathsf{PH}$ does not collapse, or assuming any other reasonable assumption in structural complexity, what hard-on-average $\mathsf{NP}$ problems are proven not to lie in $\mathsf{Average\mbox{-}P/poly}$ (if any)?
A definition of $\mathsf{Average\mbox{-}P/poly}$ can be found in Relations between Average-case and Worst-case Complexity. Thanks to Tsuyoshi for pointing out that I actually need to use $\mathsf{Average\mbox{-}P/poly}$ instead of $\mathsf{P/poly}$.
I think there are problems such as (the decision versions of) FACTORING or DLOG which are conjectured to lie in $\mathsf{NP} - \mathsf{Average\mbox{-}P/poly}$, but the conjecture is not proven based on separations between complexity classes. (Please correct me if I'm wrong.)