All closed hyperbolic 3-manifold groups embed into a compact Lie group.
To prove this, note first of all that given a hyperbolic 3-manifold $M$, it suffices to show that a finite-index subgroup $G\leq \pi_1(M)$ of index $m$ embeds into a compact Lie group. Then the representation $\rho: G\hookrightarrow O(n)$ will induce a represenation $Ind_G^{\pi_1(M)} \rho : \pi_1(M) \hookrightarrow O(nm)$.
Now, by the proof of the virtual Haken conjecture, $\pi_1(M)$ has a finite-index subgroup $G$ which is the fundamental group of a special cube complex, which implies that $G$ embeds into a right-angled artin group $A$, and hence into a right-angled Coxeter group $\Gamma$.
Finally, right-angled Coxeter groups have faithful embeddings into $O(n)$. This follows from a result of Vinberg, which gives a faithful linear action by reflections on $\mathbb{R}^n$. We'll review this construction following section 7 of this paper.
Fix a right-angled Coxeter group $$\Gamma = \langle \gamma_1,\dots,\gamma_k ~|~ (\gamma_i \gamma_j)^{m_{i,j}} = 1\quad \forall i,j\rangle,$$ where $m_{i,i}=1$ and $m_{i,j}\in\{ 2,\infty\}$ for all $i\neq j$.
For $t \in \mathbb{R}$, the matrix $M_t=(M_t(i,j))_{1\leq i,j\leq k}$ where $$M_t(i,j) = \left\{ \begin{array}{cl} 1 & \text{if }m_{i,j}=1, \text{ i.e. $i=j$},\\ 0 & \text{if }m_{i,j}=2,\\ -t & \text{if }m_{i,j}=\infty \end{array}\right.$$ defines a symmetric bilinear form $\langle\cdot,\cdot\rangle_t$ on $\mathbb{R}^k$. Note that $\mathrm{det}(M_t)$ is a nonzero polynomial in $t$ (take $t=0$), hence it is nonzero outside of some finite set $F$ of exceptional values of $t$. For any $t\in \mathbb{R}-F$, the form $\langle\cdot,\cdot\rangle_t$ is nondegenerate. Define the representation $\rho_t : \Gamma\to\mathrm{Aut}(\langle\cdot,\cdot\rangle_t)\leq \rm{GL}(k,\mathbb{R})$ by $$\rho_t(\gamma_i) : v \mapsto v - 2\langle v, e_i \rangle_t \, e_i $$ for all $i$. Each generator is a reflection in a hyperplane perpendicular to $e_i$ with respect to the metric $\langle\cdot,\cdot\rangle_t$.
For $t>1$, the convex cone $$\widetilde{\Delta}_t = \{ v\in\mathbb{R}^k ~|~ \langle v, e_i\rangle_t \leq 0 \ \,\forall i\}$$ descends to a convex polytope $\Delta_t$ in an affine chart of $\mathbb{P}(\mathbb{R}^{k})$. By Theorem 2 of Vinberg, the representation $\rho_t$ is discrete, faithful, the set $\rho_t(\Gamma)\cdot\Delta_t$ is convex in $\mathbb{P}(\mathbb{R}^k)$, and the action of $\Gamma$ on the open set $$\mathcal{U}_t:=\mathrm{Int} \left ( \rho_t(\Gamma)\cdot\Delta_t \right )$$ is properly discontinuous.
Now, let $t$ be close to $0$ and transcendental, so that $M_t$ is positive definite. Then $\rho_t : \Gamma\to \mathrm{Aut}(\langle\cdot,\cdot\rangle_t) \cong O(k)$ is faithful.