In Bar-Natan's paper "On the Vassiliev Knot Invariants", he says that it is not difficult to extend the definition of Vassiliev invariants for (unframed) knots to framed knots, but I don't know how to extend it.
If we define it in the most obvious way, then I think we can't assure that every ($F$-valued) Vassiliev invariant (of degree $m$) gives a $F$-valued function on chord diagrams (of degree $m$). It is because that when $K$ is a framed singular knot of degree $m+1$ and $K_+$(resp. $K_-$) is its positive (resp. negative) desingularization with respect to one singular point, then the framing of $K_+$ is twisted two more than the framing of $K_-$.
A possible modification of the definition I came up with is to change the way of extending framed knot invariant $V$ to framed singular knots. Specifically, instead of using $V(K)=V(K_+)-V(K_-)$, we use $V(K)=V(K_+)-V(K_-')$ where $K_-'$ is a framed singular knot whose underlying singular knot is the same as $K_-$ and whose framing is one more than the framing of $K_-$. However this modification fails because $F$-valued functions on chord diagrams coming from "framed Vassiliev invariants" may not be weight systems (i.e. may not satisfy $4T^*$-relation).
So how should we define Vassiliev invariants for framed oriented knots?