I assume you mean that $T$ acts preserving each fiber. Then the flatness says that the multigraded Hilbert polynomial is constant. As the Duistermaat-Heckman measure is the leading-order behavior of the multigraded Hilbert polynomial, it too is constant. Then the moment polytope is the support of the DH measure, so it, too, is constant.

Here's an example to consider: $F = Proj\ \mathbb C[x^1,y^1,z^1,\epsilon^0]/\langle xz - \epsilon y^2\rangle$ over $Spec\ \mathbb C[\epsilon]$, where the exponents indicate the degree, and $\mathbb C^\times$ acts on $x,y,z,\epsilon$ with weights $0,1,2,0$ respectively. Then for $\epsilon\neq 0$, the $\epsilon$-fiber is a smooth plane conic with moment polytope $[0,2]$ (bearing DH measure = Lebesgue measure). But for $\epsilon = 0$, it's a union of two lines, whose moment polytopes are individually $[0,1],[1,2]$. In particular the general fiber has two fixed points but the special fiber has three.

What's going to happen to the $1$-d orbits is that they may break (as in the example), and new ones may form (similar to the extra fixed point in the above example).

If as the question title indicates, you really want references about toric varieties, the canonical one is Bernd Sturmfels' 1990 paper on Gr\"obner bases of toric varieties.