Let $M$ be an oriented compact Riemannian manifold. Let $f$ be a Lipschitz function on $M$, denote $M'\subset M$ be the set on which $f$ is differentiable.
On one hand, Stokes theorem works for Lipschitz functions, so we have $$0=\int_M \Delta f.$$
On the other hand I was wondering do we have $$\int_M\Delta f=\int_{M'}\Delta f\ \ ?$$
In other words, if $f$ is Lipschitz on $M$, and $\Delta f\geq 0$ on $M'$, could we conclude that $f$ has to be a constant?
If $f$ is only Lipschitz continuous, then $\Delta f$ is not in general an $L^2$ function. Instead, it has to be interpreted as a distribution. (For example, on $\mathbb R$, the Laplacian of $|x|$ is a delta distribution concentrated at the origin.) It follows easily from the definition of distributional derivatives that $\int_M\Delta f=0$ for any $L^2$ function (indeed, any distribution) $f$.
When integrating a distribution, it's definitely not OK to throw out sets of measure zero. (Think about the delta distribution, for example.)
To address your main question, I think it might be true that if $f$ is Lipschitz and $\Delta f\ge 0$ in the distributional sense (which means that $\int_M \phi\Delta f \ge 0$ for any smooth nonnegative test function $\phi$), then $f$ is constant. But I don't have time to look it up right now.