It is easier to start looking at $\mathbb{R}$ and then generalize.
When considering the case in $\mathbb{R}$, let us first look at something easier than $C_0^\infty$. Put $u_n=\frac{1}{n^2}\chi_{[-n,n]}$, then $\int u_n dx=\int_{-n}^n\frac{1}{n^2}dx=\frac{2}{n}\to0$ Now choose $v$ that blows up at $\infty$ ("faster than $n$") e.g. $v=\sum_1^\infty n^2 u_n$ since compact sets are bounded they live in an some bounded interval so $v\in L^1_\text{loc}$ and $\int u_n v =\int_{-n}^n\frac{1}{n^2}v=\frac{1}{n^2}\int_{-n}^n\sum_0^nk^2 u_kdx=\\\frac{1}{n^2}\sum_1^n\int_{-k}^kk^2 \frac{1}{k^2}dx=\frac{n(n+1)}{n^2}\to1\not=0.$ Surely, we may choose smooth $u_n$ using e.g. through a convolution or redefining them at the boundaries.
When this is done, we can consider the case in $\mathbb{R^3}$ using balls instead of intervals or just put $u_n(x,y,z)=u_n(x)u_n(y)u_n(z)$.
Edit: The above is for $p=1$, for $p>1$ may look at consider $U_n=u_n^{1/p}$ and $V=v^{1/p}$.