There are several serious obstructions. One is provided by Bell's analysis concerning the conflict between realism and locality. However that obstruction concerns a very peculiar situation, referring to a bipartite system, with parts causally separated, and quantum entangled states.
There is another no-go result, usually called Kochen-Specker theorem, leading to a very severe obstruction against any completely classical interpretation of Quantum Mechanics based on hidden variables and epistemic randomness (however Bohmian quantum mechanics is untouched by it).
Actually this theorem exists into a number of versions and its origin can be traced back to the celebrated Gleason's theorem as observed by Bell himself in his second famous paper of 1966 preceding the paper by Kochen and Specker of 1967.
The basic idea underpinning the no-go result is that quantum observables $A$ (selfadjoint operators on the Hilbert space of the system) are actually classical variables and there is a classical hidden state $\lambda$ (a set of hidden classical variables $\lambda \in \Lambda$) which fixes the values $v_\lambda(A) \in \mathbb R$ of every observable $A$.
In this view, randomness of values attained by measurements of quantum observables is explained by assuming that $\lambda$ is unknown, but we know only a probability distribution $\mu$ over $\Lambda$ describing the probability that $\lambda$ attains some value (discrete distribution) or stay in some "continuous" set. This is what happens, for instance, in classical statistical mechanics. Here quantum probability becomes epistemic instead of ontic as in the standard interpretation of QM. In other words there must exist some correspondence $\mu \leftrightarrow |\psi \rangle $ such that
$$\langle \psi| A \psi \rangle = \int_{\Lambda} v_\lambda(A) d\mu(\lambda)\:.$$
It remains to fix general rules to associate sharp values $v_\lambda(A)$ to observables $A$. The problem is how one should deal with functional relations as $C=A+B$. The naive idea to always assume that $v_\lambda(C) = v_\lambda(A) + v_\lambda(B)$ turns out to be untenable when $A$ and $B$ are quantistically described as incompatible observables as explained by Bell analysing an earlier no-go theorem by von Neumann in 1966.
A fair set of assumptions for $A \mapsto v_\lambda(A)$, which avoids to tackle any classical interpretation of quantum incompatibility, was proposed by Kochen and Specker referring to the algebra of observables $B(\cal H)_{sa}$ over a finite-dimensional Hilbert space $\cal H$ (finite dimensionality requirement can be relaxed by assuming some suitable continuity requirement on $v_\lambda$).
(1) The map $v_\lambda : B({\cal H})_{sa} \ni A \mapsto v_\lambda(A) \in \mathbb R $ is non-trivial (not all values are $0$).
(2) If $A,B \in B(\cal H)_{sa}$ are compatible observables (i.e. they commute), then
$v_{\lambda}(A+B) = v_\lambda(A)+ v_\lambda(B)$.
(2) If $A,B \in B(\cal H)_{sa}$ are compatible observables (i.e. they commute), then
$v_{\lambda}(AB) = v_\lambda(A)v_\lambda(B)$.
A more precise theory would also fix how the map $v_\lambda$ deals with incompatible observables. This thechnical specification is not necessary for producing the no-go result I go to state and this fact also shows how KS' result is powerful.
Kochen-Specker Theorem
If $3\leq \dim(\cal H) < +\infty$, then there is no map $v_\lambda : B(\cal H)_{sa} \ni A \mapsto v_\lambda(A) \in \mathbb R $ satisfying requirements (1), (2), (3).
This theorem rules out from scratch every classical interpretation of QM where the realism hypothesis, i.e. every quantum observable is actually classical and always has an (unknown) sharp value. All that before any attempt to explain quantum randomness in terms of some classical uncertainty.
Actually, a closer scrutiny shows that there is a way out when assuming the contextuality requirement: that the always existing values $v_\lambda(A)$ depend also on which observable $B$ I measure together with $A$ ($B$ is therefore assumed to be compatible with $A$). It may happens that $v_\lambda(A|B)\neq v_\lambda(A|B')$ if $B$ and $B'$ are incompatible (compatibility is not a transitive relation!). This impervious approach seems to be logically consistent even if it requires a big revision of our classical ideas on the physical world (personally I definitely prefer the standard interpretation of QM!).
The result of Kochen and Specker rules out realistic non-contextual hidden-variable interpretations of quantum theory.
There is an equivalent formulation of K-S theorem which is more suitable for experiments. It is based on the notion of test. A test is an observable which can assume only the value $0$ or $1$, in the standard formalism tests are all of orthogonal projectors $P \in B(\cal H)_{sa}$.
Kochen-Specker Theorem
If $3\leq \dim(\cal H) < +\infty$, then there is a set $\cal P$ of tests such that there is no map $v_\lambda: {\cal P} \ni P \to \{0,1\}$ satisfying the following requirements
(1) If $P,P' \in \cal P$ are compatible mutually exclusive tests ($PP'=0$ as orthogonal prjectors), then at most one of $v_\lambda(P)$, $v_\lambda(P')$ does not vanish.
(2) If $P_1,\ldots, P_n \in \cal P$ is a set of pairwise compatible and mutually exclusive tests such that $P_1+\ldots + P_n =I$, then one of $v_\lambda(P_k)$ does not vanish.
The original proof of KS theorem in 1967 proved that if $\dim(\cal H)=3$ there is a set of 117 tests satisfying the theorem. Actually a general proof valid for every dimension (also infinite when assuming some continuity hypothesis on $v_\lambda$) easily arises from Gleason's theorem as already noticed by Bell.
I have seen that deterministic QM interpretations like Bohmian mechanics introduce concepts like infinite dimensional configuration space.
I do not think so. Bohmian mechanics for particles is formulated in the standard $3N$ dimensional configuration space of a system of $N$ particle. Maybe you are considering the system of a quantum field. I am not an expert on this subject however.
As recent references I would like to mention various entries of Stanford Encyclopedia of Philosophy, Landsman's book on foundations of quantum theory, a book consisting of a wide collections of recent papers on Bell's analysis and further foundational issues.
(I am publishing a book on fundamental mathematical structures in quantum theoryand chapter 5 is completely devoted to study these issues including Bell's inequality and its interplay with locality and contextuality).
My answer here could be of interest
