Is an ample divisor linearly equivalent to a non-trivial effective divisor In other words, if $L$ is an ample divisor, must we have $h^0(L)>0$ or can this be zero?

An ample divisor need not have global sections.

To see this, first note that any divisor of positive degree on a curve is ample. On the other hand, if $C$ is a curve of genus 2, and $p,q,r$ are general points on $C$, then the line bundle $\mathcal{O}_C(p+q-r)$ has no global sections.