  Let $I=I(D)$ be the edge ideal of a weighted oriented graph $D$, let $G$ be
the underlying graph of $D$, and let $I^{(n)}$ be the $n$-th symbolic power of
$I$ defined using the minimal primes of $I$. We prove that $I^2=I^{(2)}$ if and
only if (i) every vertex of $D$ with weight greater than $1$ is a sink and (ii)
$G$ has no triangles. As a consequence, using a result of Mandal and Pradhan,
and the classification of normally torsion-free edge ideals of graphs, it
follows that $I^n=I^{(n)}$ for all $n\geq 1$ if and only if (a) every vertex of
$D$ with weight greater than $1$ is a sink and (b) $G$ is bipartite. If $I$ has
no embedded primes, conditions (a) and (b) classify when $I$ is normally
torsion-free. Using polyhedral geometry and integral closure, we give necessary
conditions for the equality of ordinary and symbolic powers of monomial ideals
with a minimal irreducible decomposition. Then, we classify when the Alexander
dual of the edge ideal of a weighted oriented graph is normally torsion-free.
