Once more, with feeling. Thanks to comments from Tyler Lawson and Neil Strickland.

Let me use $B_n$ for the finite Boolean lattice of subsets of $[n] := \{1,2,\ldots,n\}$ (your $\mathcal{P}(S)$), and let $k=|T|$. Then your $A=((B_k\setminus \{\hat{1}\}) \times B_{n-k}) \setminus \{\hat{0}\}$, where $\hat{0}$ means the minimum of a poset, and $\hat{1}$ the maximum. To understand the homotopy type of this product, we need a result of Quillen (stated as Theorem 5.1(b) in the paper "Canonical homeomorphisms of posets" by Walker cited below; see also the discussion in Section 5.1 of the notes of Wachs), which says that
$$ |P\times Q \setminus \{\hat{0}\}| \simeq |P\setminus \{\hat{0}\}| \ast |Q\setminus\{\hat{0}\}|,$$
for posets $P,Q$ which have minimums, where $\ast$ is the join of complexes. So in our case
$$ |A| \simeq |B_k \setminus \{\hat{0},\hat{1}\}| \ast |B_{n-k}\setminus \{\hat{0}\}|.$$
But $B_{n-k}\setminus \{\hat{0}\}$ is contractible (it has a maximum), so $|A|$ is also contractible.

Note that if we considered the poset $B=\{U\subseteq [n]\colon U \cap T \neq \varnothing, T \not\subseteq U\}$ as in the answer of Neil Strickland, then we have that $B = (B_k \setminus \{\hat{0},\hat{1}\}) \times B_{n-k}$. [And this was the poset that my first answer posted here was implicitly about.] Using the another fact about homotopy types of products (see Theorem 5.1(a) in the paper of Walker) that 
$$|P\times Q| \simeq |P| \times |Q|$$ 
(product of complexes) means that 
$$|B| \simeq |B_k \setminus \{\hat{0},\hat{1}\}| \times |B_{n-k}|,$$ and since $B_{n-k}$ is contractible, this time we actually can say $|B| \simeq |B_k \setminus \{\hat{0},\hat{1}\}| \simeq \mathbb{S}^{k-2}$.

<cite authors="Quillen, Daniel">_Quillen, Daniel_, [**Homotopy properties of the poset of nontrivial p-subgroups of a group**](https://doi.org/10.1016/0001-8708(78)90058-0), Adv. Math. 28, 101-128 (1978). [ZBL0388.55007](https://zbmath.org/?q=an:0388.55007).</cite>

<cite authors="Wachs, Michelle L.">_Wachs, Michelle L._, [**Poset topology: tools and applications**](https://arxiv.org/abs/math/0602226), Miller, Ezra (ed.) et al., Geometric combinatorics. Providence, RI: American Mathematical Society (AMS); Princeton, NJ: Institute for Advanced Studies (ISBN 978-0-8218-3736-8/hbk). IAS/Park City Mathematics Series 13, 497-615 (2007). [ZBL1135.06001](https://zbmath.org/?q=an:1135.06001).</cite>

<cite authors="Walker, James W.">_Walker, James W._, [**Canonical homeomorphisms of posets**](https://doi.org/10.1016/S0195-6698(88)80033-7), Eur. J. Comb. 9, No. 2, 97-107 (1988). [ZBL0661.06006](https://zbmath.org/?q=an:0661.06006).</cite>