I don't think you can generally conclude the nonnegativity of Ehrhart polynomials when expressed in the standard monomial basis; however, the more natural basis for Ehrhart polynomials is given by the binomial coefficients $\binom{n+d} d$, $\binom{n+d-1} d$, ..., $\binom n d$ (here $n$ is the variable of the polynomial and $d$ is the dimension of $P$). Stanley proved in 1980 that the coefficients of the Ehrhart polynomial of $P$ written in terms of this basis are nonnegative. This coefficient vector is known as the $h^*$-vector, $\delta$-vector, or Ehrhart $h$-vector of $P$. Here are some things that are known and connected to your question:

If $P$ has a unimodular triangulation $T$ then the $h^*$-vector of $P$ equals the $h$-vector of $T$ (a result contained in the aforementioned 1980 paper by Stanley). (I don't think a pulling triangulation buys anything extra, short of nice properties of the $h$-vector.)

For the case that $P$ is compressed, I believe the state of the art is described in a 2007 paper by Bruns and Roemer in JCTA.

For the case that $P$ is integrally closed, you may consult a recent preprint by Braun and Davis.