For any poset $P$, we can define an abstract simplicial complex $\Delta{P}$ called its order complex, whose vertices are the elements of $P$ and whose faces are chains (totally ordered subsets) of $P$. The geometric realisation $\lvert \Delta{P} \rvert$ of this complex is a topological space. This is described in [1, p. 6].
A topological space $(X,T)$ has a homotopy class $[T]$.
As $T$ is a poset, we may also take the realization of the order complex of $T$, $S = \lvert \Delta(\mathcal{T}) \rvert$, and this has a homotopy class $[S]$.
Are $[S]$ and $[T]$ equal? If not, what is the relationship between them? I'm especially interested in the case when $(X,T)$ is a finite $\mathrm{T}_0$ space.
[1] Wachs, Michelle L. "Poset topology: tools and applications." arXiv preprint math/0602226 (2006). https://arxiv.org/pdf/math/0602226.pdf
The order complex of any poset $P$ with a greatest or least element $x$ is contractible. Indeed, $|\Delta P|$ is the cone on $|\Delta (P\setminus\{x\})|$: each simplex in $\Delta(P\setminus \{x \})$ (i.e., chain in $P\setminus\{x\}$) becomes a cone over that simplex with vertex $x$ (since you can add $x$ as an extra point to the chain), and it can be checked that these glue together to give a cone over the entire space $|\Delta (P\setminus\{x\})|$. (There are other ways to see this as well; for instance, thinking of $P$ as a category, then $x$ is an initial or terminal object of $P$, and so gives an adjunction between $P$ and the terminal category which becomes a homotopy equivalence after taking nerves.)
So in particular, if $T$ is a topology, then $|\Delta T|$ is always contractible, and in particular is typically not (weakly) homotopy equivalent to the original space $X$.
(What is true is that if $X$ is a finite topological space or more generally an Alexandrov space, then $X$ is canonically weakly homomtopy equivalent to $|\Delta X|$ where $X$ is ordered with the specialization preorder, i.e. the order $x\leq y$ iff $y\in\overline{\{x\}}$. This is a theorem of McCord which you can find in the paper McCord's paper Singular homology groups and homotopy groups of finite topological spaces.)