The statement that $HF^{\ast}(X,X)$ is isomorphic to $QH^\ast(X)$ is a version of the Piunikhin-Salamon-Schwarz (PSS) isomorphism (proved, under certain assumptions, in McDuff-Salamon's book "J-holomorphic curves in symplectic topology"). PSS is a canonical ring isomorphism from $QH^{\ast}(X)$ to the Hamiltonian Floer cohomology of $X$, and the latter can be compared straightforwardly to the Lagrangian Floer cohomology of the diagonal.
Now to Hochschild cohomology of the Fukaya category $F(X)$. There's a geometrically-defined map $QH^{\ast}(X) \to HH^{\ast}(F(X))$, due to Seidel in a slightly different setting (see his "Fukaya categories and deformations"), inspired by the slightly vague but prescient remarks of Kontsevich from 1994. One could define this map without too much trouble, say, for monotone manifolds. It's constructed via moduli spaces of pseudo-holomorphic polygons subject to Lagrangian boundary conditions, with an incidence condition of an interior marked point with chosen cycles in $X$. The question is whether this is an isomorphism.
This statement is open, and will probably not be proven true in the near future, for a simple reason: $QH^*(X)$ is non-trivial, while we have no general construction of Floer-theoretically essential Lagrangians.
There are two positive things I can say. One is that Kontsevich's heuristics, which involve interpreting $HH^{\ast}$ as deformations of the identity functor, now have a natural setting in the quilted Floer theory of Mau-Wehrheim-Woodward (in progress). This says that the Fukaya category $F(X\times X)$ naturally embeds into the $A_\infty$-category of $A_\infty$-endofunctors of $F(X)$.
The other is that for Weinstein manifolds (a class of exact symplectic manifolds with contact type boundary), there seems to be an analogous map from the symplectic cohomology $SH^{\ast}(X)$ (a version of Hamiltonian Floer cohomology on the conical completion of $X$) to $HH^{\ast}$ of the wrapped Fukaya category, which involves non-compact Lagrangians. (Edit August 2010: I was careless about homology versus cohomology. I should have said that $HH_{\ast}$ maps to $SH^{\ast}$.) Proving that this is an isomorphism is more feasible because one may be able to prove that Weinstein manifolds admit Lefschetz fibrations. The Lefschetz thimbles are then objects in the wrapped Fukaya category.
One might then proceed as follows. The thimbles for a Lefschetz fibration should generate the triangulated envelope of the wrapped category (maybe I should split-close here; not sure) - this would be an enhancement of results from Seidel's book. Consequently, one should be able to compute $HH_{\ast}$ just in terms of $HH_{\ast}$ for the full subcategory generated by the thimbles. The latter should be related to $SH^{\ast}$ by ideas closely related to those in Seidel's paper "Symplectic homology as Hochschild homology".
What could be simpler?
ADDED: Kevin asks for evidence for or against $QH^{\ast}\to HH^{\ast}$ being an isomorphism. I don't know any evidence contra. Verifying it for a given $X$ would presumably go in two steps: (i) identify generators for the (triangulated envelope of) $F(X)$, and (ii) show that the map from $QH^{\ast}$ to $HH^{\ast}$ for the full subcategory that they generate is an isomorphism. There's been lots of progress on (i), less on (ii), though the case of toric Fanos has been studied by Fukaya-Oh-Ohta-Ono, and in this case mirror symmetry makes predictions for (i) which I expect will soon be proved. In simply connected disc-cotangent bundles, the zero-section generates, and both $HH_{\ast}$ for the compact Fukaya category and $SH^{\ast}$ are isomorphic to loop-space homology, but I don't think it's known that the resulting isomorphism is Seidel's.
Added August 2010: Abouzaid (1001.4593) has made major progress in this area.
I noticed this question has been bumped up to the front page, and the
most recent answer is about 8 years old: the subject has moved on
since then, and more has been written. Here is my understanding of
some of the recent developments.
Caveat: This whole area has undergone rapid progress in the last few
years, and I am not working on this question, so what I say is
probably not up-to-date (even as I write, never mind in the future).
Kontsevich's idea was that you can relate the Fukaya category of a
Weinstein manifold to the microlocal sheaf category of its
skeleton. As far as I know, a proof of this conjecture is work in
progress by Ganatra-Pardon-Shende (one preliminary part of which is
already available).
I think the idea of their proof is roughly the following:
You prove that the Fukaya category has a co-sheaf property, which
means it can be computed locally first on some subsets and then the
answer can be glued together using homotopy colimits. For this, you
need suitable functors relating the Fukaya category of a subset to
the Fukaya category of the whole manifold. This is complicated by
the fact that your "subsets" might not be very nicely embedded in the
whole manifold: for example, if your ambient manifold is $T^*M$ then
you want to allow subsets like $T^*M'$ where $M'\subset M$ is a
codimension zero submanifold with boundary (that's because you
ultimately want to work locally on the skeleton). That is
problematic because the Liouville vector field for $T^*M'$ and the
Liouville vector field for $TM$ don't match up nicely. The first GPS
paper constructs these categories and functors for "Liouville
sectors" (a suitably broad class of inclusions, related to Sylvan's
notion of stops and partially wrapped Floer homology). I think the proof of the co-sheaf property is still ongoing work?
Now you compute the local pieces of the Fukaya category and show
that they agree with the microlocal sheaf categories; since both
have co-sheaf gluing, you get the same global answers.
The second part relies on some local computations of Fukaya
categories. Nadler has introduced the
notion of "arboreal skeleton" which is a skeleton with certain
"generic" singularities. For example, trivalent graphs in dimension 1;
trivalent graph times interval or cone on 1-skeleton of a tetrahedron
in dimension 2; etc. He computes the microlocal sheaf category for
these; I'm not sure if the corresponding partially-wrapped Fukaya
categories have been calculated in all cases yet. Finally, you want to
show that any Weinstein manifold has an arboreal skeleton:
Starkston has some results in this
direction which may represent the state of the art.
Leaving this aside for a moment, there are also special cases where
the Konstevich conjecture/local-to-global results for the Fukaya
category has been established independently of this general
program. These include (but again, I'm probably missing some):
Best Answer
At the risk of writing things that are obvious to those listening in: this is Nadler-land, no?
If $X$ is a smooth complex variety with reductive group $G$ acting, and $\mu_{\mathbb C}: T^*X\rightarrow {\mathfrak g}^*$ is the complex moment map, then $\mu_{\mathbb C}^{-1}(0)/G = T^*(X/G)$ provided one interprets all quotients as stacks.
If $T^*X$ is hyperkahler and we do the hyperkahler quotient for the maximal compact of $G$, picking a nontrivial real moment value $\mu_{\mathbb R}^{-1}(\zeta)$ at which to reduce amounts (by Kirwan) to imposing a GIT stability on $\mu_{\mathbb C}^{-1}(0)$---i.e. to picking a nice open subset of the cotangent stack $T^*(X/G)$ that is actually a variety. A stack version of Nadler's "microlocal branes" theorem would describe the (suitable, undoubtedly homotopical/derived) exact Fukaya category as the constructible derived category of $X/G$.
Since I'm completely ignorant of how the Nadler-Zaslow/Nadler story actually works, I'd like to then imagine that such an equivalence microlocalizes properly to give an equivalence over the hyperkahler reduction (i.e. the nice open set) as well? Admittedly, by microlocalizing to the stable locus one should avoid all the derived unpleasantness (this should be analogous to what happens in Bezrukavnikov-Braverman's proof of "generic" geometric Langlands for $GL_n$ in characteristic $p$, where by localizing to the generic locus, ${\mathcal D}$-module really means ${\mathcal D}$-module, not "module over the enveloping algebroid of the tangent complex" or something like that).
Admittedly, I don't have a clue how to deal with the issue that the base $X$ in the important examples is typically affine...maybe if one forces some kind of boundary conditions also in the $X$-direction one could make the Fukaya category nontrivial in Tim's example of the Hilbert scheme??