Already the fact that $u$ is a tempered distribution and is weakly harmonic implies that $u$ is a polynomial. Then observe that the only polynomial in $L^p({\mathbb R}^n)$ is zero. 

Another way is to note that weakly harmonic functions are smooth (in fact analytic), hence harmonic. In particular, you do not need to talk about "almost everywhere". Then use Liouville's theorem.