Find a function which is harmonic on the area bounded by positive x axis and the line $y=x$, which goes to $0$ on the boundary, which is not identically $0$. Why doesn't it violate the max/min modulus principle?
I know that the function should be inside the "triangle", I was thinking about using the definition of harmonic functions, so my guess is that the function has to have a degree $\leq 2$.
Since the function has to be $0$ when $x=y$ or $x\in \mathbb R$ I tried to come up with examples like $x=y/2$ which decreases and goes to $0$ if we go up or down to the real line.
I'm not clear how to make it to go to $0$ on the boundary.
Can you do the same for the upper-half plane? ($u(x,y) = y = \operatorname{Im}(z)$ would be a good choice).
The function $z \mapsto z^4$ maps your "triangle" onto the upper-half plane, which suggests that $$ u(x,y) = \operatorname{Im}(z^4) = 4xy(x^2-y^2) $$ works. Verify that it does.
As for the question about the maximum principle, check the conditions in the theorem.