As Misha mentions, his paper with Leeb, "Actions of discrete groups on nonpositively curved spaces" MR1411351, gives severe restrictions on CAT(0) structures on $MCG(S)$, which covers the case of lattice embeddings of $MCG(S)$ into a semisimple Lie group $G$, and more general Lie groups are then easily handled (@Misha I think this is what your comment refers to, please correct if not). This generalizes in a couple of directions. Kida's theorem, in ``Measure equivalence rigidity of the mapping class group'' MR2680399, puts severe restrictions on any embedding of the mapping class group $MCG(S)$ of a surface $S$ as a lattice in a locally compact, second countable group $G$. And as a consequence of the quasi-isometric rigidity theorem for $MCG(S)$ of Behrstock, Kleiner, Minsky and myself, MR2928983 "Geometry and rigidity of mapping class groups", one gets severe restrictions on embeddings of $MCG(S)$ as a cocompact discrete subgroup of a compactly generated group $G$. In each case, the limitations are similar: roughly speaking, $G$ differs from $MCG(S)$ itself only by compact or bounded normal subgroups and finite index stuff. The arguments apply as well to embeddings of finite index subgroups of $MCG(S)$.
Let me give a careful statement and proof for the case I know best, which uses our QI-rigidity theorem cited above, namely the case where $MCG(S)$ embeds as a cocompact discrete subgroup of a compactly generated group $G$. The conclusion is that there is a retraction homomorphism $\rho : G \to MCG(S)$ with bounded kernel (I think the kernel must also be compact open, which is true when $G$ is a Lie group, but I am not sure in general). I'll restrict to the case that $S$ is not a 2-holed torus because our QI-rigidity theorem is slightly convoluted in that case. Using the compactly generated word metric on $G$, cocompactness implies that there is a "closest point retraction" $p : G \to MCG(S)$ which moves points a bounded distance $\le A$, and so $p$ is a $1,2A$ quasi-isometry. For each $g \in G$ consider the left multiplication map $L_g : G \to G$ which is an isometry with respect to the compactly generated word metric. Precompose $L_g$ with the inclusion $i : MCG(S) \hookrightarrow G$ which is a $K,C$ quasi-isometry for some $K,C$, and postcompose with the closest point retraction $p$, and the resulting map $p \circ L_g \circ i : MCG(S) \to MCG(S)$ is a $K,C'$ quasi-isometry for some $K,C'$ independent of $g$. Write this map as $\phi \mapsto p(g \cdot \phi)$. By the QI-rigidity theorem cited above, there exists a constant $B$ independent of $g$ and there exists a unique $\rho(g) \in MCG(S)$, such that $d(p(g \cdot \phi),\rho(g) \cdot \phi) \le B$ for all $\phi \in MCG(S)$. This function $\rho$ is a retraction because if $g$ is already in $MCG(S)$ then the inequality $d(g \cdot \phi, \rho(g) \cdot \phi) < A$ for all $\phi$ implies $g=\rho(g)$, by uniqueness. A similar line of thought using uniqueness shows that $\rho$ is a homomorphism. The kernel of $\rho$ is bounded because, by the triangle inequality, if $\rho(g) = Id$ then the quantity $d(Id,g) = d(\rho(g),g) = d(\rho(g) \cdot Id,g \cdot Id)$ differs from the quantity $d(p(g \cdot Id),\rho(g) \cdot Id) \le B$ by at most $d(g \cdot Id,p(g \cdot Id)) = d(g,p(g)) \le A$.