In working with a problem where I keep getting results that go against my basic understanding of algebraic geometry, I figure that it might be worthwhile to check that that basic understanding of algebraic geometry is right in the first place. Thus, here we are.
Let $R_1 , R_2 , R_3$ be the coordinate rings of varieties $V_1 , V_2 , V_3$ respectively. Is it then always the case that if $R_1 \subseteq R_2 \subseteq R_3$ then $V_3 \subseteq V_2 \subseteq V_1$?
Not necessarily. Consider $k[x_1]\subset k[x_1,x_2]\subset k[x_1,x_2,x_3]$. Then $V_i = \Bbb A^i_k$, and the claimed relation does not hold.
Instead, what is true is that the inclusion maps $R_i\subset R_{i+1}$ give maps $\operatorname{Spec} R_{i+1} \to \operatorname{Spec} R_i$, and many different things may happen here - the basic issue is that just being a ring extension doesn't force many things to happen without other conditions. If you fix a base ring $R$ and ideals $I_1\subset I_2\subset I_3$, then it is true that $V(I_3)\subset V(I_2) \subset V(I_1)$ by the Nullstellensatz.