In the statement of the Baire Category Theorem, one needs to include openness of a set so that the theorem holds.
Question: what is the example such that countable intersection of dense sets is not dense?
EDIT: perhaps I should rephrase my question:
Give an example such that countable intersection of dense sets is not dense and nonempty.
Even a finite intersection need not be dense. In the reals, $\mathbb{Q}$ and $\mathbb{R}\setminus\mathbb{Q}$ are both dense, and have empty intersection.
If we have finitely many dense and open subsets $D_1,\ldots,D_N$, in any space $D_1 \cap \ldots \cap D_N$ is (open) and dense. This is a motivation for Baire spaces: here we can go to countable intersections, not just finite ones. The intersection need not be open anymore, but is still dense in Baire spaces.
To get a non-empty intersection of two, add $(0,1)$ to both the rationals and the irrationals. The intersection is $(0,1)$, not dense.
Give an example such that countable intersection of dense sets is not dense and nonempty.
It's easy to modify the given counterexample of $\Bbb Q$ and $\Bbb R \backslash \Bbb Q$ to match your needs. For example, you can take,
$$ \Bbb Q \cup \{\pi\}, \ \Bbb Q \cup \{\pi, \pi^2\}, \ \Bbb Q \cup \{\pi, \pi^2, \pi^3 \}, \ \dots, $$
these sets are all dense in $\Bbb R,$ but their intersection in $\{\pi\}.$
In fact, you can prove a stronger result, namely that there exists a countable collection of dense subsets of $\Bbb R$ whose intersection is pairwise not dense and non-empty. Indeed, if $p_1, p_2, \dots$ is an enumeration of the primes, then the collection, $$\ (\Bbb Q + \sqrt{p_1}) \cup \{\pi\}, \ (\Bbb Q + \sqrt{p_2}) \cup \{\pi\}, \ (\Bbb Q + \sqrt{p_3}) \cup \{\pi\}, \ \dots, $$ where $\Bbb Q + \sqrt{p_i} = \{ \sqrt{p_i} + q \mid q \in \Bbb Q \}.$ It can be shown that these sets are dense in $\Bbb R,$ but the intersection of any subcollection is $\{\pi\}$ (this isn't exactly easy to prove, but see this post for one).
