I appreciate the comments so far and am modifying based on something closer to the problem I'm interested in. I started out with something far too general.

This is probably easy, but I have been away from math for quite some time now so what I'm asking may be ancient history by now. Anyway, I'm wondering what conditions are needed for the following to be true.

Suppose $Y = \cap H_i$ is a sub-scheme of a normal variety $U$, let $\overline{Y}$ be the closure in some $X$, and let $D = X{\setminus} U$. Then is $\overline{Y} = \cap \overline{H_i}$ if $\overline{Y} \cap D = \overline{H_i} \cap D$? If not, can we place conditions on $D$ such as integral, normal, Cartier (or maybe all three?), and/or can we place conditions on $Y$ such as reduced or integral?

For example, I was hoping the twisted cubic $C \subset X =\mathbb{P}^3$ would yield a counter-example, but if we take $U = \mathbb{A}^3$, $D$ the line at infinity and two hypersurfaces which cut out $C$ on $\mathbb{A}$ but only set-theoretically cut out $C$ globally, we'll get a point of multiplicity $3=deg(C)$ on $C\cap D$ but a point of multiplicity 6 on on the intersection of $D$ with the two hypersurfaces (the point at infinity is a double point on the intersection of the two hypersurfaces).

The reason I am interested in this is is the following. Let $I$ be an ideal in $k[M]$, $M{\cong}{\mathbb{Z}^n},$ $w{\in}{N{=}M^{*}},$ and $I_w{=}k[M_w]{\cap}I$ where $k[M_w]$ is the subring generated by monomials with positive $w$-weight. Then $I_w$ is the ideal of the closure of $V(I)$ in the toric variety $U_w = \text{Spec}(k[M_w])$. One can check that generators of the initial ideal $\text{in}{_w}I$ (here we take the terms of minimal weight) generate the ideal of both $V(I)$ and $\overline{V(I)}\cap O$ where $O$ is the closed orbit in $U_w$.

So, if my question is true, a set $\{f_1, \ldots, f_r\} \subset I$ produce generators for $\text{in}_wI$ when closures of the associated hypersurfaces cut-out the closure of $V(I)$. This would be another example of how the tropicalization reflects the asymptotic behavior of a subvariety in the torus - to find a tropical basis for $I$, one should look for hypersurfaces which cut out $Y$ not in $T$ but in some larger toric variety.