I can't really answer this. I'll just think out loud for a bit.
Let $R$ be a complete local $\mathbb{F}_p$ algebra. The additive formal group $G$ is a formal scheme $Spf(\mathbb{F}_p[[x]])$. An $R$-point of $G(R)$ is an element of $\mathfrak{m}_R$.
Pick $t\in G(R)=\mathfrak{m}_R$, and consider $f(x)=x(x-t)(x-2t)\cdots (x-(p-1)t)=x^p-tx$. Let $F_t=Spec(R[[x]]/(x^p-tx))=Spec(R[x]/(x^p-tx))$. Then $F_t$ is a finite subgroup scheme of $G$. If we base change $F_t$ to $\tilde{F}_t$ over $R[t^{-1}]$, then $\tilde{F}_t$ becomes an etale group scheme.
The universal example of such an $F_t$ lives over $B=\mathbb{F}_p[[t]]$. The scheme $S$ of automorphisms of $G$ (i.e., the dual Steenrod algebra) acts on $B$, and the action lifts to $B[t^{-1}]=\mathbb{F}_p((t))$. If $\omega$ is the module of invariant differentials on $G$ (isomorphic as a module with $S$-action to $tB/t^2B$, then there's a map
$$Res_{t=0}: B[t^{-1}] \otimes_{\mathbb{F}_p} \omega\to \mathbb{F}_p,$$
which is a map of $S$-modules. Lin's theorem asserts that this map induces isomorphisms in $Ext_S^*(\omega^i,{-})$.
So Lin's theorem is something about residues.
You have this residue map is other cases, for instance if we replace $G$ with a Lubin-Tate deformation. Neil Strickland has thought about this: in his Formal Schemes and Formal Groups, he spells out some of the relationship between the residues and the Segal conjecture.
For a general audience it is much better to treat $MO$ rather than $MU$, because the complex orientation creates many unpleasant subtleties. (See papers of Buchstaber and Ray for interesting examples where these subtleties make a concrete and computable difference.) Let us say that a geometric chain of dimension n in X is an equivalence class of pairs $(M,f)$, where M is a compact smooth manifold (possibly with boundary) and $f:M\to X$ is a continuous map. Here $(M,f)$ is equivalent to $(M',f')$ if there is a diffeomorphism $u:M\to M'$ with $f'u=f$. We write $GC_{\ast}(X)$ for the graded abelian monoid of geometric chains. This has a differential $\partial[M,f]=[\partial M,f|_{\partial M}]$, and the homology is $MO_{\ast}(X)$. (This needs a few remarks about homology of complexes of monoids, but there is nothing very subtle going on.) I have a general audience talk about $MO$ at http://neil-strickland.staff.shef.ac.uk/talks/durham.pdf
Best Answer
As far as I know, there is still no such interpretation. The closest I've heard is some rumored (but unpublished) work in derived algebraic geometry interpreting MU as some kind of representing object.
Such a construction of MU in terms of formal group data be very welcome (probably even more now than when Ravenel wrote the green book).
EDIT: Some elaboration.
We do know a lot about MU. We know that it has an orientation (Chern classes for vector bundles), and in it's universal for this property. It's not then extremely suprising that we get a formal group law from the tensor product for line bundles, but the fact that MU carries a universal formal group law, and that MU ^ MU carries a universal pair of isomorphic formal group laws, is surprising. At this point it's something we observe algebraically. Even Lurie's definition of derived formal group laws, assuming I understand correctly, is geared to construct formal group laws objects in derived algebraic geometry carrying a connection to the formal group law data that we already know is there on the spectrum level, and hence ties it to the story we already knew for MU implicitly.
Some reasons these days we might want to know how to construct MU from formal group law data: