As alluded to in a comment of @PietroMajer the reason that one can't solve in the space of compactly supported functions is somewhat subtle (and in particular, I think the other answers are not complete).

Basically, for a solution of $\Delta_g f=g$ to exist with both $f,g\in C^\infty_c(M)$ one has to have that $g$ is $L^2$ orthogonal to every harmonic function.  There are, in general, a plethora of these when the ambient space is non-compact and these impose serious restrictions on the allowed $g$.  

Just to give a concrete example in $\mathbb{R}^2$.  Let $\phi\geq 0$ be a non-negative compactly supported function with $\phi=1$ on $B_1$.  Set $g=(x^2-y^2)\phi$.  Now suppose one had a compactly supported solution $f$ to $\Delta f=g$.  Integration by parts gives
\begin{align*}
0&=\int_{\mathbb{R}^2} f \Delta (x^2-y^2) \\
&=\int_{\mathbb{R}^2} (x^2-y^2) \Delta f \\
&= \int_{\mathbb{R}^2} \phi (x^2-y^2)^2 >0
\end{align*}
This is a contradiction so there can be no such solution.

It might also be illustrative to see what happens in $\mathbb{R}$.  You can see that to solve $f''=g$ for $f$ and $g$ compactly supported as this is just an integration and the conditions are quite explicit.