The boundary points of union of two sets is a subset of boundary points of 1st set union boundary points of second set. That is if $A$ is topology on $X$ then $$Fr(A\cup B) \subseteq Fr(A) \cup Fr(B)$$
I need a counter example in which $Fr(A\cup B)$ is not a subset of $Fr(A) \cup Fr(B)$
If $Fr(A\cup B) \subseteq Fr(A)\cup Fr(B)$ is always true, then you can't find an example where that doesn't happen.
I suppose you are asking the oposite: an example where $Fr(A)\cup Fr(B)$ is not a subset of $Fr(A\cup B)$.
In that case in $\mathbb{R}$, pick $A = (1,3)$ and $B = (2,4)$.
We have $Fr(A) = \{1,3\}$ and $Fr(B) = \{2,4\}$
So $Fr(A)\cup Fr(B) = \{1,2,3,4\}$ which is not a subset of $Fr(A\cup B) = \{1,4\}$