What are some open problems in toric varieties? In light of the nice responses to this question, I wonder what are some open problems in
the area of toric geometry?  In particular, 

What are some open problems relating to the algebraic combinatorics of toric varieties?

and

What are some open problems relating to the algebraic geometry of toric varieties?

 A: The following question was posed by Rikard Bögvad in the paper On the homogeneous ideal of a projective nonsingular toric variety:

Is the toric ideal of a smooth
  projectively normal toric variety
  generated by quadrics?

This is interesting, since toric ideals have an explicit description. In particular, it is not known if the coordinate ring of a smooth projectively normal toric variety is Koszul. Smoothness is of course essential here, since there are many toric hypersurfaces of degree $\ge 3$, e.g., $x_0^n=x_1 \cdots x_n$.
A: My favourite is Oda's Strong Factorization Conjecture:

Can a proper, birational map between smooth toric varieties be factored as a composition of a sequence of smooth toric blow-ups followed by a sequence smooth toric blow-downs?

Note that if you are allowed to intermingle the blow-ups and blow-downs (the weak version) it has been proved.  In fact, it was proved for general varieties in characteristic 0 using the toric case:
Torification and Factorization of Birational Maps. Abramovich, Karu, Matsuki, Wlodarczyk.
A conjectural algorithm for computing toric strong factorizations can be found in the following arXiv article:
On Oda's Strong Factorization Conjecture. Da Silva, Karu.
