I think the answer is "no". As I suggested in a now-deleted answer, we can restrict ourselves to considering the restriction to a torus invariant $\mathbb{P}^1$. A line bundle on a toric variety is determined by its restriction to each such $\mathbb{P}^1$, and is anti-ample if and only if its restriction to such a $\mathbb{P}^1$ is.

Let $P$ be a torus invariant $\mathbb{P}^1$. Let the normal bundle of $P$ be $\bigoplus_{i=1}^{n-1} \mathcal{O}(a_i)$. So there is a torus invariant open neighborhood of $P$ which has two affine charts: The first one is $\mathrm{Spec} \ k[t, x_1, \ldots, x_{n-1}]$ and the second is $\mathrm{Spec} \ k[u, y_1, \ldots, y_{n-1}]$ where, on the overlap, we have $t=u^{-1}$ and $y_i = t^{-a_i} x_i$. (I am using that $X$ is smooth. Otherwise, the normal bundle might not be a vector bundle, and I have to work harder.)

Let your line bundle $L$, restricted to $P$, be $\mathcal{O}(-b)$. Since your line bundle is assumed anti-ample, we have $b >0$. Let $q$ and $r$ be generators of your line bundle in the two charts, so $r = t^{-b} q$. (I really hope I'm getting all the signs right. Someone please check!)

We now want to consider the Frobenius pushforward, $F_* L$. This has rank $p^n$. In the first chart, a basis is $ t^{c} \prod x_i^{k_i} q$ for $0 \leq c, \ k_i < p$. In the second chart, a basis is $ u^{d} \prod y_i^{\ell_i} r$ with similar bounds. And $$u^{d} \prod y_i^{\ell_i} r = t^{-b} t^{-d} \prod t^{-a_i \ell_i} x_i^{\ell_i} q = (t^p)^{-N} t^c \prod x_i^{\ell_i} q$$ where $c \equiv -b-d - \sum a_i \ell_i \mod p$, with $0 \leq c < p$, and $N = \left( b+c+d + \sum a_i \ell_i \right)/p$. We want to show that $N > 0$. The first three terms are positive, but there is plenty of room for the last term to be negative.

In particular, look at the toric surface whose fan has rays in directions $(1,0)$, $(-1, 1)$, $(-1, 0)$, $(-1, -1)$. The four boundary curves have self-intersection, respectively, $(2,0,-2,0)$. Let $L$ be the line bundle whose restrictions to these curves has degree $(-3,-1,-1,-1)$. Let $P$ be the boundary $\mathbb{P}^1$ with self-intersection $-2$, so $a_1=-2$. Then the restriction of $F_* L$ to $P$ has degrees $$\frac{-1+c+d - 2 \ell}{p}$$ where $0 \leq d, \ell < p$ and $c$ is chosen in that same range in order to make the quotient an integer.

This is $-1$ whenever $2 (\ell/p) - (d/p) > 1$, which happens for about $1/4$ of the possible values of $(c , \ell)$.

I think the answer is "no". As I suggested in a now-deleted answer, we can restrict ourselves to considering the restriction to a torus invariant $\mathbb{P}^1$. A line bundle on a toric variety is determined by its restriction to each such $\mathbb{P}^1$, and is anti-ample if and only if its restriction to such a $\mathbb{P}^1$ is.

Let $P$ be a torus invariant $\mathbb{P}^1$. Let the normal bundle of $P$ be $\bigoplus_{i=1}^{n-1} \mathcal{O}(a_i)$. So there is a torus invariant open neighborhood of $P$ which has two affine charts: The first one is $\mathrm{Spec} \ k[t, x_1, \ldots, x_{n-1}]$ and the second is $\mathrm{Spec} \ k[u, y_1, \ldots, y_{n-1}]$ where, on the overlap, we have $t=u^{-1}$ and $y_i = t^{-a_i} x_i$. (I am using that $X$ is smooth. Otherwise, the normal bundle might not be a vector bundle, and I have to work harder.)

Let your line bundle $L$, restricted to $P$, be $\mathcal{O}(-b)$. Since your line bundle is assumed anti-ample, we have $b >0$. Let $q$ and $r$ be generators of your line bundle in the two charts, so $r = t^{-b} q$. (I really hope I'm getting all the signs right. Someone please check!)

We now want to consider the Frobenius pushforward, $F_* L$. This has rank $p^n$. In the first chart, a basis is $ t^{c} \prod x_i^{k_i} q$ for $0 \leq c, \ k_i < p$. In the second chart, a basis is $ u^{d} \prod y_i^{\ell_i} r$ with similar bounds. And $$u^{d} \prod y_i^{\ell_i} r = t^{-b} t^{-d} \prod t^{-a_i \ell_i} x_i^{\ell_i} q = (t^p)^{-N} t^c \prod x_i^{\ell_i} q$$ where $c \equiv -b-d - \sum a_i \ell_i \mod p$, with $0 \leq c < p$, and $N = \left( b+c+d + \sum a_i \ell_i \right)/p$. In other words, $N = \lceil (b+d+\sum a_i \ell_i)/p \rceil$ where $\lceil x \rceil$ is the "round up" function. Our bundle is anti-ample if $N > 0$. The first three terms are positive, but there is plenty of room for the last term to be negative.

In particular, look at the toric surface whose fan has rays in directions $(1,0)$, $(-1, 1)$, $(-1, 0)$, $(-1, -1)$. The four boundary curves have self-intersection, respectively, $(2,0,-2,0)$. Let $L$ be the line bundle whose restrictions to these curves has degree $(-3,-1,-1,-1)$. Let $P$ be the boundary $\mathbb{P}^1$ with self-intersection $-2$, so $a_1=-2$. Then the restriction of $F_* L$ to $P$ has degrees $$\lceil \frac{-1+d - 2 \ell}{p} \rceil$$ where $0 \leq d, \ell < p$.

This is $-1$ whenever $2 (\ell/p) - (d/p) > 1$, which happens for about $1/4$ of the possible values of $(c , \ell)$.

I think the answer is "no". As I suggested in a now-deleted answer, we can restrict ourselves to considering the restriction to a torus invariant $\mathbb{P}^1$. A line bundle on a toric variety is determined by its restriction to each such $\mathbb{P}^1$, and is anti-ample if and only if its restriction to such a $\mathbb{P}^1$ is.

Let $P$ be a torus invariant $\mathbb{P}^1$. Let the normal bundle of $P$ be $\bigoplus_{i=1}^{n-1} \mathcal{O}(a_i)$. So there is a torus invariant open neighborhood of $P$ which has two affine charts: The first one is $\mathrm{Spec} \ k[t, x_1, \ldots, x_{n-1}]$ and the second is $\mathrm{Spec} \ k[u, y_1, \ldots, y_{n-1}]$ where, on the overlap, we have $t=u^{-1}$ and $y_i = t^{-a_i} x_i$. (I am using that $X$ is smooth. Otherwise, the normal bundle might not be a vector bundle, and I have to work harder.)

Let your line bundle $L$, restricted to $P$, be $\mathcal{O}(-b)$. Since your line bundle is assumed anti-ample, we have $b >0$. Let $q$ and $r$ be generators of your line bundle in the two charts, so $r = t^{-b} q$. (I really hope I'm getting all the signs right. Someone please check!)

We now want to consider the Frobenius pushforward, $F_* L$. This has rank $p^n$. In the first chart, a basis is $ t^{c} \prod x_i^{k_i} q$ for $0 \leq c, \ k_i < p$. In the second chart, a basis is $ u^{d} \prod y_i^{\ell_i} r$ with similar bounds. And $$u^{d} \prod y_i^{\ell_i} r = t^{-b} t^{-d} \prod t^{-a_i \ell_i} x_i^{\ell_i} q = (t^p)^{-N} t^c \prod x_i^{\ell_i} q$$ where $c \equiv -b-d - \sum a_i \ell_i \mod p$, with $0 \leq c < p$, and $N = \left( b+c+d + \sum a_i \ell_i \right)/p$. We want to show that $N > 0$. The first three terms are positive, but there is plenty of room for the last term to be negative.

In particular, look at the toric surface whose fan has rays in directions $(1,0)$, $(-1, 1)$, $(-1, 0)$, $(-1, -1)$. The four boundary curves have self-intersection, respectively, $(2,0,-2,0)$. Let $L$ be the line bundle whose restrictions to these curves has degree $(-3,-1,-1,-1)$. Let $P$ be the boundary $\mathbb{P}^1$ with self-intersection $-2$, so $a_1=-2$. Then the restriction of $F_* L$ to $P$ has degrees $$\frac{-1+c+d - 2 \ell}{p}$$ where $0 \leq d, \ell < p$ and $c$ is chosen in that same range in order to make the quotient an integer.

This is $-1$ whenever $2 (\ell/p) - (c/p) > 1$, which happens for about $1/4$ of the possible values of $(c , \ell)$.

I think the answer is "no". As I suggested in a now-deleted answer, we can restrict ourselves to considering the restriction to a torus invariant $\mathbb{P}^1$. A line bundle on a toric variety is determined by its restriction to each such $\mathbb{P}^1$, and is anti-ample if and only if its restriction to such a $\mathbb{P}^1$ is.

Let $P$ be a torus invariant $\mathbb{P}^1$. Let the normal bundle of $P$ be $\bigoplus_{i=1}^{n-1} \mathcal{O}(a_i)$. So there is a torus invariant open neighborhood of $P$ which has two affine charts: The first one is $\mathrm{Spec} \ k[t, x_1, \ldots, x_{n-1}]$ and the second is $\mathrm{Spec} \ k[u, y_1, \ldots, y_{n-1}]$ where, on the overlap, we have $t=u^{-1}$ and $y_i = t^{-a_i} x_i$. (I am using that $X$ is smooth. Otherwise, the normal bundle might not be a vector bundle, and I have to work harder.)

Let your line bundle $L$, restricted to $P$, be $\mathcal{O}(-b)$. Since your line bundle is assumed anti-ample, we have $b >0$. Let $q$ and $r$ be generators of your line bundle in the two charts, so $r = t^{-b} q$. (I really hope I'm getting all the signs right. Someone please check!)

We now want to consider the Frobenius pushforward, $F_* L$. This has rank $p^n$. In the first chart, a basis is $ t^{c} \prod x_i^{k_i} q$ for $0 \leq c, \ k_i < p$. In the second chart, a basis is $ u^{d} \prod y_i^{\ell_i} r$ with similar bounds. And $$u^{d} \prod y_i^{\ell_i} r = t^{-b} t^{-d} \prod t^{-a_i \ell_i} x_i^{\ell_i} q = (t^p)^{-N} t^c \prod x_i^{\ell_i} q$$ where $c \equiv -b-d - \sum a_i \ell_i \mod p$, with $0 \leq c < p$, and $N = \left( b+c+d + \sum a_i \ell_i \right)/p$. We want to show that $N > 0$. The first three terms are positive, but there is plenty of room for the last term to be negative.

In particular, look at the toric surface whose fan has rays in directions $(1,0)$, $(-1, 1)$, $(-1, 0)$, $(-1, -1)$. The four boundary curves have self-intersection, respectively, $(2,0,-2,0)$. Let $L$ be the line bundle whose restrictions to these curves has degree $(-3,-1,-1,-1)$. Let $P$ be the boundary $\mathbb{P}^1$ with self-intersection $-2$, so $a_1=-2$. Then the restriction of $F_* L$ to $P$ has degrees $$\frac{-1+c+d - 2 \ell}{p}$$ where $0 \leq d, \ell < p$ and $c$ is chosen in that same range in order to make the quotient an integer.

This is $-1$ whenever $2 (\ell/p) - (d/p) > 1$, which happens for about $1/4$ of the possible values of $(c , \ell)$.