Existence of non-constant holomorphic map between two given compact Riemann surfaces
No. For example, if $f : \mathbb{CP}^1 \to \mathbb{C}/\Lambda$ is holomorphic, then because $\mathbb{CP}^1$ is simply connected, it has a lift $\tilde{f} : \mathbb{CP}^1 \to \mathbb{C}$. That is, $\tilde{f}$ satisfies $f = \pi\circ\tilde{f}$ where $\pi : \mathbb{C} \to \mathbb{C}/\Lambda$ is the universal covering map (i.e. the quotient map). Every holomorphic function on a compact complex manifold is constant, so $\tilde{f}$ is constant, and therefore so is $f$.
More generally, if $f : X \to Y$ is a non-constant holomorphic map, then $g(X) \geq g(Y)$. To see this, recall that a non-constant holomorphic map between Riemann surfaces is a branched covering. Let $f : X \to Y$ be a non-constant holomorphic map, then by the Riemann-Hurwitz formula,
$$\chi(X) = d\chi(Y) - \sum_{p \in X}(e_p - 1)$$
where $e_p$ is the ramification index at $p$ and $d \in \mathbb{Z}_+$. If $g(Y) = 0$ (i.e. $Y = \mathbb{CP}^1$), there is nothing to prove, so suppose $g(Y) > 0$. Note that
$$\chi(X) = d\chi(Y) - \sum_{p \in X}(e_p - 1) \leq d\chi(Y) \leq \chi(Y)$$
where the last inequality uses the fact that $\chi(Y) \leq 0$. Therefore, $g(X) \geq g(Y)$.
As for the existence of such maps, there are non-constant holomorphic maps $f : X \to \mathbb{CP}^1$ for any Riemann surface $X$ (of any genus); this is equivalent to the existence of a non-zero meromorphic function. I'm not sure about the case where the target has positive genus, but it is related to this question.
Let me first consider the case of nonconstant holomorphic maps $X\to T^2$, from compact connected Riemann surfaces of genus $g\ge 2$ to tori (smooth elliptic curves). Every such map is determined (up to a translation of $T^2$) by the induced map of fundamental groups $G=\pi_1(X)\to \pi_1(T^2)=Z^2$. There are only countably many homomorphisms $G\to Z^2$ and the moduli space of elliptic curves is 1-dimensional. Hence, the subset of the moduli space ${\mathcal M}_g$ consisting of Riemann surfaces admitting such holomorphic maps is a countable union of complex subvarieties of dimension $\le 1$. Since ${\mathcal M}_g$ is an irreducible variety of dimension $3g-3$, we conclude that only a meager (in the classical topology) subset of ${\mathcal M}_g$ consists of Riemann surfaces admitting non constant holomorphic maps to elliptic curves. In particular, not every Riemann surface of genus $g\ge 2$ admits such a map.
The situation is essentially the same for nonconstant holomorphic maps $X\to Y$ where $X$ and $Y$ are hyperbolic Riemann surfaces: By Severi's finiteness theorem (see my answer here), given $X$, there are only finitely many (up to a natural isomorphism) pairs $(Y, f)$ where $Y$ is a Riemann surface of genus $\ge 2$ and $f: X\to Y$ is a nonconstant holomorphic map.