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][1] 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.

  [1]: http://en.wikipedia.org/wiki/Castelnuovo%E2%80%93de_Franchis_theorem