In fact, here's an explicit example with $\chi(M)\not=0$:  Let $M$ be the 1-point compactification of the hyperelliptic Riemann surface given in the affine plane by
$$
y^2 = x^5-1.
$$
This is a smooth Reimann surface of genus $g=2$ and hence $\chi(M) = -2$.  The holomorphic differential
$$
\omega = \frac{dx}{y} = \frac{dx}{\sqrt{x^5-1}}
$$
has only one zero (over the point $p$ where $x$ and $y$ have poles of order $2$ and $5$ respectively).  Consequently, the metric
$$
g = \omega\circ\overline{\omega}
$$
is flat and is a smooth $(0,2)$ tensor that vanishes only at $p$.  

Similar hyperelliptic examples can be constructed for any genus greater than $0$.