If by "subvariety" you mean that $W$ is known to be irreducible

Let $Z$ be the base locus of $D$. The sections give a map $T \setminus Z \to \mathbb P^{11}$. Your claim about six general sections implies that $W$ does not intersect $Z$ (otherwise the intersecion with arbitrarily many sections would include $W \cap Z$ and be nonempty) and that the image of $W$ along this map has dimension at most $5$ (otherwise the intersection of this image with $6$ general hyperplanes would be nonempty). The claim about five general sections implies the image contains a $5$-dimensional component (otherwise the intersection of this image with five general hyperplanes would be empty) and that the fiber over a general point of this $5$-dimensional component is zero-dimensional (otherwise the intersection with the inverse images of five general hyperplanes would include the positive-dimensional fiber).

Certainly $W$ contains an irreducible component which maps dominantly to the $5$-dimensional component of the image of $W$ and hence is $5$-dimensional. However, if $W$ is not irreducible, there's no way to rule out that $W$ contains also a larger-dimensional component with smaller-dimensional image in $\mathbb P^{11}$.

For an explicit example, let $T$ be the blowup of a point on $\mathbb P^{11}$, which is toric. Let $D$ be a hyperplane not intersecting that point, which is nef and has a $12$-dimensional space of sections. Let $W$ be the union of a five-dimensional variety outside the exceptional divisor with an $n$-dimensionl variety inside the exceptional divisor for any $n$ from $0$ to $10$. A general section of $D$ will not intersect the excetional divisor, so the intersection with five general sections will be the same as the intersection of the five-dimensional variety with five general hyperplanes and ignore the exceptional divisor component. So this will satisfy all your hypotheses for any $n$.

If by "subvariety" you mean that $W$ is known to be irreducible, the answer is positive. Otherwise, it is negative, as I will show by an explicit counterexample.

Let $Z$ be the base locus of $D$. The sections give a map $T \setminus Z \to \mathbb P^{11}$. Your claim about six general sections implies that $W$ does not intersect $Z$ (otherwise the intersecion with arbitrarily many sections would include $W \cap Z$ and be nonempty) and that the image of $W$ along this map has dimension at most $5$ (otherwise the intersection of this image with $6$ general hyperplanes would be nonempty). The claim about five general sections implies the image contains a $5$-dimensional component (otherwise the intersection of this image with five general hyperplanes would be empty) and that the fiber over a general point of this $5$-dimensional component is zero-dimensional (otherwise the intersection with the inverse images of five general hyperplanes would include the positive-dimensional fiber).

Certainly $W$ contains an irreducible component which maps dominantly to the $5$-dimensional component of the image of $W$ and hence is $5$-dimensional. 

However, if $W$ is not irreducible, there's no way to rule out that $W$ contains also a larger-dimensional component with smaller-dimensional image in $\mathbb P^{11}$.

For an explicit example, let $T$ be the blowup of a point on $\mathbb P^{11}$, which is toric. Let $D$ be a hyperplane not intersecting that point, which is nef and has a $12$-dimensional space of sections. Let $W$ be the union of a five-dimensional variety outside the exceptional divisor with an $n$-dimensionl variety inside the exceptional divisor for any $n$ from $0$ to $10$. A general section of $D$ will not intersect the excetional divisor, so the intersection with five general sections will be the same as the intersection of the five-dimensional variety with five general hyperplanes and ignore the exceptional divisor component. So this will satisfy all your hypotheses for any $n$.

If by "subvariety" you mean that $W$ is known to be irreducible

Let $Z$ be the base locus of $D$. The sections give a map $T \setminus Z \to \mathbb P^{11}$. Your claim about six general sections implies that $W$ does not intersect $Z$ (otherwise the intersecion with arbitrarily many sections would include $W \cap Z$ and be nonempty) and that the image of $W$ along this map has dimension at most $5$ (otherwise the intersection of this image with $6$ general hyperplanes would be nonempty). The claim about five general sections implies the image contains a $5$-dimensional component (otherwise the intersection of this image with five general hyperplanes would be empty) and that the fiber over a general point of this $5$-dimensional component is zero-dimensional (otherwise the intersection with the inverse images of five general hyperplanes would include the positive-dimensional fiber).

Certainly $W$ contains an irreducible component which maps dominantly to the $5$-dimensional component of the image of $W$ and hence is $5$-dimensional. However, if $W$ is not irreducible, there's no way to rule out that $W$ contains also a larger-dimensional component with smaller-dimensional image in $\mathbb P^{11}$.

If by "subvariety" you mean that $W$ is known to be irreducible

Let $Z$ be the base locus of $D$. The sections give a map $T \setminus Z \to \mathbb P^{11}$. Your claim about six general sections implies that $W$ does not intersect $Z$ (otherwise the intersecion with arbitrarily many sections would include $W \cap Z$ and be nonempty) and that the image of $W$ along this map has dimension at most $5$ (otherwise the intersection of this image with $6$ general hyperplanes would be nonempty). The claim about five general sections implies the image contains a $5$-dimensional component (otherwise the intersection of this image with five general hyperplanes would be empty) and that the fiber over a general point of this $5$-dimensional component is zero-dimensional (otherwise the intersection with the inverse images of five general hyperplanes would include the positive-dimensional fiber).

Certainly $W$ contains an irreducible component which maps dominantly to the $5$-dimensional component of the image of $W$ and hence is $5$-dimensional. However, if $W$ is not irreducible, there's no way to rule out that $W$ contains also a larger-dimensional component with smaller-dimensional image in $\mathbb P^{11}$.

For an explicit example, let $T$ be the blowup of a point on $\mathbb P^{11}$, which is toric. Let $D$ be a hyperplane not intersecting that point, which is nef and has a $12$-dimensional space of sections. Let $W$ be the union of a five-dimensional variety outside the exceptional divisor with an $n$-dimensionl variety inside the exceptional divisor for any $n$ from $0$ to $10$. A general section of $D$ will not intersect the excetional divisor, so the intersection with five general sections will be the same as the intersection of the five-dimensional variety with five general hyperplanes and ignore the exceptional divisor component. So this will satisfy all your hypotheses for any $n$.