Let $(M,g)$ be a three-dimensional non-compact Riemannian manifold without boundary (oriented and connected).
Does there exist a (smooth, global) vector field $X\in\mathfrak{X}(\Sigma)$ such that the Lie derivative $\mathcal{L}_{X}$ viewed as a map $\mathcal{L}_{X}:C_{c}^{\infty}(\Sigma)\to C^{\infty}_{c}(\Sigma)$ is injective? (The compact support here is crucial in my question).
If the above is not true in general, can we maybe at least formulate some geometrical or topological (e.g. homological) conditions on $(M,g)$ such that it is true, i.e. such that there exists such a vector field?
Note that your question is independent of the metric $g$.
Here is a partial answer (correcting the previous answer). Suppose $M$ admits a vector field so that for a dense set of points $D\subset M$ the closure of the flow $\overline{\{\Phi^t(x)\mid t\in\Bbb R\}}$ is not compact for all $x\in D$ (and the flow is taken for all times at which it is defined). Then $X:C^\infty_c(M)\to C^\infty_c(M)$ is injective.
To see it let $g\in C_c^\infty(M)$ with $X(g)=0$. Then for all points $x\in M$ you have that $\frac{d}{dt}g(\Phi^t(x))=X(g)(\Phi^t(x))=0$ and $g$ is constant along the flow of $X$. In particular if $x\in D$ and $g(x)\neq0$ then $g(y)\neq0$ for all $y$ in some closed but not compact set. This implies that $g$ cannot have compact support, so if $g$ has compact support then $g(x)=0$ for all $x\in D$, which was dense.