Does $\operatorname{int} A \subseteq \operatorname{int} B$ imply $\partial A \subseteq \partial B$?
Let $X$ be a topological space and $A,B \subseteq X$. Clearly, $\operatorname{int} A \subseteq \operatorname{int} B$ does not imply $A \subseteq B$. A counter-example is $A = (1, 2]$ and $B = [1, 2)$.
Does $\operatorname{int} A \subseteq \operatorname{int} B$ imply $\partial A \subseteq \partial B$?
No, consider the example $A=[1,2]$, $B=(0,3)$, then we have that $\text{int}(A)=(1,2)$, $\text{int}(B)=(0,3)$, and $\partial A=\{1,2\}$, and $\partial B=\{0,3\}$.
Let $A=(0,1)$, and let $B=(0,2)$. Then, $\partial A=\{0,1\}\not\subseteq\{0,2\}=\partial B$.