A few opposite-looking remarks, and the case of (minimal) surfaces.

If a variety $X$ admits a non-constant morphism to a curve, then it admits a non-constant morphism to $\mathbb{P}^1$ (just compose the given morphism with a morphism of the curve to $\mathbb{P}^1$). Thus, if you only care about existence of a non-constant morphism to a curve, you may as well restrict your attention to the question of existence of a non-constant morphism to $\mathbb{P}^1$. On the other hand, the existence of a morphism to a curve of genus at least one is a birational property: any such morphism factors through the Albanese variety of $X$, and the Albanese variety of $X$ is a birational invariant. The property of admitting a morphism to $\mathbb{P}^1$ is clearly not a birational invariant property, as any non-constant rational function determines a rational map to $\mathbb{P}^1$ (see Charles Matthews' answer).

Both questions appear quite hard, though. An equivalent formulation of the question is the following: does $X$ admit two disjoint effective non-zero nef divisors? The equivalence of the statements is almost tautological. An easy implication is that the rank of the Neron-Severi group of $X$ is at least two, and that there are effective non-zero nef divisors that are not big.

For minimal surfaces, the situation is as follows. A surface of negative Kodaira dimension (i.e. a ruled surface or $\mathbb{P}^2$, here we do not need the surface to be minimal) admits a morphism to $\mathbb{P}^1$ if and only if it is not isomorphic to $\mathbb{P}^2$. A surface of Kodaira dimension zero (i.e. a K3, Enriques, Abelian of bielliptic surface) not admitting a morphism to $\mathbb{P}^1$ is a non-elliptic K3 surface. Every surface of Kodaira dimension one (a properly elliptic surface) admits a morphism to $\mathbb{P}^1$ (and in fact a canonical morphism to a curve).

For all surfaces (including the surfaces of Kodaira dimension two), one thing you can say is that there is a Theorem of Castelnuovo and de Franchis characterizing surfaces with a morphism to a curve of genus at least two.

Thus, in conclusion, it seems that for minimal surfaces of special type, the only surfaces not admitting a morphism to a curve are $\mathbb{P}^2$ and non-elliptic K3 surfaces (which does not look so bad, after all!). Note that every irreducible component of the moduli space of polarized K3 surfaces contains elliptic surfaces and non-elliptic ones.

A few opposite-looking remarks, and the case of (minimal) surfaces.

If a variety $X$ admits a non-constant morphism to a curve, then it admits a non-constant morphism to $\mathbb{P}^1$ (just compose the given morphism with a morphism of the curve to $\mathbb{P}^1$). Thus, if you only care about existence of a non-constant morphism to a curve, you may as well restrict your attention to the question of existence of a non-constant morphism to $\mathbb{P}^1$. On the other hand, the existence of a morphism to a curve of genus at least one is a birational property: any such morphism factors through the Albanese variety of $X$, and the Albanese variety of $X$ is a birational invariant. The property of admitting a morphism to $\mathbb{P}^1$ is clearly not a birational invariant property, as any non-constant rational function determines a rational map to $\mathbb{P}^1$ (see Charles Matthews' answer).

Both questions appear quite hard, though. An equivalent formulation of the question is the following: does $X$ admit two disjoint effective non-zero nef divisors? The equivalence of the statements is almost tautological. An easy implication is that the rank of the Neron-Severi group of $X$ is at least two, and that there are effective non-zero nef divisors that are not big.

For minimal surfaces, the situation is as follows. A surface of negative Kodaira dimension (i.e. a ruled surface or $\mathbb{P}^2$, here we do not need the surface to be minimal) admits a morphism to $\mathbb{P}^1$ if and only if it is not isomorphic to $\mathbb{P}^2$. A surface of Kodaira dimension zero (i.e. a K3, Enriques, Abelian of bielliptic surface) not admitting a morphism to $\mathbb{P}^1$ is a non-elliptic K3 surface. Every surface of Kodaira dimension one (a properly elliptic surface) admits a morphism to $\mathbb{P}^1$ (and in fact a canonical morphism to a curve).

EDIT: Among surfaces of Kodaira dimension zero, also simple abelian surfaces (i.e. abelian surfaces that are not isogenous to a product of two elliptic curves) admit no morphism to a curve.

For all surfaces (including the surfaces of Kodaira dimension two), one thing you can say is that there is a Theorem of Castelnuovo and de Franchis characterizing surfaces with a morphism to a curve of genus at least two.

Thus, in conclusion, it seems that for minimal surfaces of special type, the only surfaces not admitting a morphism to a curve are $\mathbb{P}^2$ and non-elliptic K3 surfaces and simple abelian surfaces (which does not look so bad, after all!). Note that every irreducible component of the moduli space of polarized K3 surfaces contains elliptic surfaces and non-elliptic ones.