If $\Omega$ is not all $\mathbb{R}^{n}$, the function $$\int_{\Omega}f(x-y)\phi(y)dy$$
is not well defined for all $x$, so im gonna assume that $\Omega=\mathbb{R}^{n}$.
Let $F(x,y)=f(x-y)\phi(y)$. Note that \begin{eqnarray}
\int_{\mathbb{R}^{n}}|F(x,y)|dx &=& \int_{\mathbb{R}^{n}}|f(x-y)|\phi(y)|dx \nonumber \\
&=& |\phi(y)|\|f\|_{L^{1}(\mathbb{R}^{n})}
\end{eqnarray}
Hence, $$\int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}}|F(x,y)|dxdy=\|\phi\|_{L^{1}(\mathbb{R}^{n})}\|f\|_{L^{1}(\mathbb{R}^{n})}
$$
By Tonelli's theorem (see Brezis - Functional Analyis, Sobolev Spaces and PDE, page 91) $F\in L^{1}(\mathbb{R}^{n}\times\mathbb{R}^{n})$.
Then by Fubini's theorem (see Brezis - Functional Analyis, Sobolev Spaces and PDE, page 91), we have for all $\phi\in C^{\infty}_{C}(\mathbb{R}^{n})$
\begin{eqnarray}
\int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}}f(x-y)\phi(y)dydx &=& \int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}}f(x-y)\phi(y)dxdy \nonumber \\
&=& \|\phi\|_{L^{1}(\mathbb{R}^{n})}\|f\|_{L^{1}(\mathbb{R}^{n})}\\
&=& 0
\end{eqnarray}
Therefore, $f=0$.