Here, $T(F)$ is statistical functional of F such that $T(F) = F(b) - F(a)$ for fixed number $a$ and $b$. We have a ECDF $\widehat{F}_n$ for $X_1, ....X_n$ from distribution $F$. Our task is to find $Var(\widehat{θ})$ where $ \widehat{θ} = T(\widehat{F}_n) = \widehat{F}_n(b) - \widehat{F}_n(a)$.
First check whether $$ \mathbb E\hat\theta=\theta $$ Use the definition of empirical distribution function $$ \mathbb E[\hat F(t)]=\mathbb E\left[\frac1n \sum_{i=1}^n \mathbb 1_{\{X_i\leq t\}}\right] $$ and the linearity property of expectation.
For calculating variance check and use the identity $$ \mathbb 1_{\{X_i\,\leq\, b\}} - \mathbb 1_{\{X_i\,\leq\, a\}} = \mathbb 1_{\{a\,<\,X_i\,\leq \,b\}} \quad \text{for} \quad a<b. $$ and variance properties: $\mathop{\text{Var}}(cX)=c^2\mathop{\text{Var}}(X)$. $$ \mathop{\text{Var}}(\hat\theta) = \mathop{\text{Var}}\left(\frac1n \sum_{i=1}^n \mathbb 1_{\{X_i\leq b\}}-\frac1n \sum_{i=1}^n \mathbb 1_{\{X_i\leq a\}}\right) = \frac1{n^2}\mathop{\text{Var}}\,\left(\sum_{i=1}^n \mathbb 1_{\{a<X_i\leq b\}} \right). $$ Find the distribution of a sum in brackets and its variance.