Given symmetric $A\in\mathbb{R}^{m\times m}$, $B\in\mathbb{R}^{m\times n}$, symmetric $D\in\mathbb{R}^{n\times n}$, assume the eigenvalues $\lambda_i\in [\alpha_4,\alpha_3]\times[-\beta_2,\beta_2]\bigcup[\alpha_2,\alpha_1]\times[-\beta_1,\beta_1]$ of the eigenvalue problem:
$$ \begin{equation} \begin{bmatrix} A & B\\ B^T & D \end{bmatrix}x = \lambda x, \end{equation} $$
for $\alpha_4<\alpha_3<\gamma<0<\Gamma<\alpha_2<\alpha_1$ and $0<\beta_1,\beta_2$. Consider the eigenvalue problem:
$$ \begin{equation} (D - B^TA^{-1}B)x = \mu x. \end{equation} $$
We know that $\mu_i\neq 0$ due to the Haynsworth inertia additivity formula, but can there still be $\mu_i = a+bi$ and $\gamma\leq a\leq\Gamma$? (Equivalently, can the eigenvalues of the Schur complement be arbitrarily close to zero if the eigenvalues of the original matrix are bounded away from zero?)
Edit: Fixed a typo in the statement of the second eigenvalue problem.
I figured it out and I am leaving this answer, in case someone(me) needs in the future.
First of all, if $A$ and $D$ are symmetric and real then $\beta_1=\beta_2=0$, which is one of the cases I am interested in. More generally, given $A$ is real but not symmetric then it may be possible to analyze the eigenvalues by separating $A$ in to its symmetric and skew-symmetric parts and apply the following argument. However, I am just going to discuss the answer for my original question.
Define $$M = \begin{bmatrix} A & B\\B^T & D \end{bmatrix}.$$ By Lemma 2.3 in the book "The Schur complement and its applications*", eigenvalues of the inverse the Schur complement $(D-B^TA^{-1}B)^{-1}$ interlaces with the eigenvalues of the inverse $M^{-1}$. In rigorous terms, $\lambda_i(M^{-1})\leq \lambda_i((D-B^TA^{-1}B)^{-1})\leq\lambda_{i+k}(M^{-1})$, where $k$ is the size of $D-B^TA^{-1}B$ and eigenvalues are ordered so that $\lambda_i\leq\lambda_j$ if $i<j$. Remind that, $\lambda_i(M^{-1}) = 1/\lambda_i(M)$. Now, we need to consider few cases $|\gamma|<1$ or $|\gamma|\ge 1$ and $\Gamma<1$ or $\Gamma\ge 1$. I am going to skip this part as it is straightforward and long. The idea is that we find a negative value $c$ such that $1/\mu_i>c$ and a positive value $C$ such that $1/\mu_i<C$. In the end, we can conclude that $\mu$ will be bounded away from zero by $\gamma$ and $\Gamma$. This tells us that the eigenvalues of the Schur complement are not closer to zero than the eigenvalues of the original system. However, they may be far far away due to the interlacing of the inverses.
* Note that, this book is not the main source for the some of the interlacing theorems of Chapter 2. Nevertheless, it is a comprehensive source. If you are looking to cite any of these theorems, IMO, the best practice is to cite the original source in addition to this book.