To get such a result we typically need that the degenerate subspaces include via cofibrations, and we can get a counterexample by picking a standard non-cofibration.
Let $X_0 = \{0\}$, and let $X_1 = \{0, 1, 1/2, 1/3, \ldots\} \subset \Bbb R$, with degeneracy $s^0: X_0 \to X_1$ being the natural inclusion. Build the rest of $X$ so that all higher simplices are degenerate: let $X_p = X_1 \vee \cdots \vee X_1$ with $p$ factors. The face maps are collapsing down a factor on the side, or using the fold map on two adjacent factors; the degeneracies are obtained by inserting a $0$.
Let $Y$ be the simplicial space with the same underlying set, but where we've now given everything the discrete topology. The natural map $Y \to X$ is a levelwise equivalence because both are totally disconnected.
Both simplicial spaces have only a single $0$-simplex, some space of $1$-simplices, and all higher simplices are degenerate. The geometric realization of such a $Z$ is the space
$$
\frac{Z_1 \times [0,1]}{Z_1 \times \{0,1\} \cup \{*\} \times [0,1]} = Z \wedge S^1.
$$
Thus, the map $|Y| \to |X|$ is the standard continuous bijection from a countable wedge of circles to $\{0,1,1/2,1/3,\ldots\} \wedge S^1$, which is homeomorphic to the Hawaiian earring (via the standard "continuous bijection from compact to Hausdorff" argument).
Vidit, thanks for the advertisement; Paul I'll answer your email shortly.
As a minor point, there is a small but subtle mistake in Clader's work
that is corrected in Matthew Thibault's 2013 Chicago thesis, which goes
further in that direction.
I do intend to finish the advertised book, but it is too incomplete to
circulate yet. There is actually a large and interesting picture that
connects mainstream algbraic topology to combinatorics via finite spaces.
However, the right level
of generality is $T_0$-Alexandroff spaces, $A$-spaces for short. These
are topological spaces in which arbitrary rather than just finite
intersections of open sets are open, and of course finite $T_0$-spaces
are the obvious examples. One can in principle answer Paul's question in the
affirmative, but the finiteness restriction feels artificial and the connection
between $A$-spaces and simplicial complexes is far too close to ignore.
The category of $A$-spaces is isomorphic to the category of posets, $A$-spaces
naturally give rise to ordered simplicial complexes (the order complex of a
poset) and thus to simplicial sets, while abstract simplicial complexes naturally
give rise to $A$-spaces (the face poset).
Subdivision is central to the theory, and barycentric subdivision of a
poset is WHE to the face poset of its order complex.
Categories connect up since the second subdivision of a category is
a poset, which helps illuminate Thomason's equivalence between the
homotopy categories of $\mathcal{C}at$ and $s\mathcal{S}et$.
Weak and actual homotopy equivalences are wildly different for
$A$-spaces. In the usual world of spaces, they correspond to
homotopy equivalences and simple homotopy equivalences, respectively,
a point of view that Barmak's book focuses on.
The $n$-sphere is WHE to a space with $2n+2$ points, and that is the
minimum number possible.
If the poset $\mathcal{A}_pG$ of non-trivial elementary abelian $p$-subgroups of a
finite group $G$ is contractible, then $G$ has a normal $p$-subgroup. A celebrated conjecture of Quillen says in this language that if
$\mathcal{A}_pG$ is weakly contractible (WHE to a point), then it is
contractible and hence $G$ has a normal $p$-subgroup. There are many
interesting contractible finite spaces that are not weakly contractible.
These facts just scratch the surface and were nearly all previously known,
but there is much that is new in the book, some of it due to students
at Chicago where I have taught this material in our REU off and on since 2003.
This is ideal material for the purpose. (Obsolete notes and even current ones
can be found on my web page by those sufficiently interested to search: Minian,
Barmak's thesis advisor in Buenos Aires, found them there and started off work
in Argentina based on them.) I apologize for this extended advertisement,
but perhaps Paul's question gives me a reasonable excuse.
Best Answer
To avoid leaving this question open:
Assuming we work in the category of compactly generated spaces, geometric realization commutes with pullbacks.(It's crucial that we use the compactly generated product.) The proof is basically the same as for simplicial sets. A reference in the space-case is Corollary 11.6 of Peter May's book 'The Geometry of Iterated Loop Space'. The terminal object is preserved as well, so the claim follows by abstract nonsense, since a functor preserves all finite limits iff it preserves pullbacks and the terminal object.