Having two closed exact Lagrangian submanifolds $L_1$ and $L_2$ that intersect cleanly inside a Liouville manifold $M$ with $c_1(M)=0,$ is there (with possibly some other conditions) any relation between $$H^*(L_1 \cap L_2)$$ and $$HF^*(L_1,L_2)?$$ In particular, (when) are they necessarily isomorphic?
There is a spectral sequence from the cohomology of the intersection to the Floer cohomology. This is explained (and used) in Seidel's paper "Lagrangian 2-spheres can be symplectically knotted"
and is based on the thesis of Pozniak, available here:
(As far as I remember, Pozniak didn't mention the spectral sequence, but it's one way of understanding his work).
Edit: In particular, to answer your last question, if the Lagrangians are exact and the intersection is connected then the spectral sequence degenerates at $E_2$ and the Floer cohomology is isomorphic to the cohomology of the intersection. If there are J-holomorphic discs (e.g. potentially when the Lagrangians are non-exact) then this might not be true. The example to bear in mind is $L_1=L_2$: the intersection is connected, and we have $HF(L,L)=H(L)$ whenever L is exact, but not in general.