  We study the base locus of the higher fundamental forms of a projective toric
variety $X$ at a general point. More precisely we consider the closure $X$ of
the image of a map $({\mathbb C}^*)^k\to {\mathbb P}^n$, sending $t$ to the
vector of Laurent monomials with exponents $p_0,\dots,p_n\in {\mathbb Z}^k$. We
prove that the $m$-th fundamental form of such an $X$ at a general point has
non empty base locus if and only if the points $p_i$ lie on a suitable
degree-$m$ affine hypersurface.
  We then restrict to the case in which the points $p_i$ are all the lattice
points of a lattice polytope and we give some applications of the above result.
In particular we provide a classification for the second fundamental forms on
toric surfaces, and we also give some new examples of weighted $3$-dimensional
projective spaces whose blowing up at a general point is not Mori dream.
