*The following is a rephrasing of and an expansion on Joris Bierkens' answer. It serves as my note for my own benefit. It would be a bonus if it helps any other people.*

Define stochastic processes $\displaystyle y_t := \exp\Big(-\int_0^t r_s \,ds \Big)$. The subscript denote the time variable dependency. Write $f(t) := \mathbb E [y_t]$. We apply the Ito's lemma recursively. In fact, we can derive the Ito version of the Taylor expansion to an arbitrary order.

Apply Ito's Lemma twice
\begin{align}
-(y_t-y_0) &=\int_0^tr_sy_sds \\
&=\int_0^tds\Big(r_0y_0+\int_0^s(y_udr_u+r_udy_u)\Big) \\
&=r_0t+\int_0^tdu\ y_u(\mu_u-r_u^2)(t-u)+\int_0^tdB_u\,y_u\sigma_u(t-u)
\end{align}
since $dr_tdy_t=-r_ty_tdr_tdt=0$. The last integral comes from exchanging the order of two integrations. We in essence have a Ito version of the Taylor expansion up to the first order with remainders.
Take expectation of the above expression. As $\mathbb E \left[ (r_t^2 - \mu_t) y_t \right]$ is bounded,
$$f(t) = 1-r_0t+O(t^2)=\exp\big(-r_0t+O(t^2)\big)$$
as $t\searrow 0$.

It remains to fill in details regarding the interchange of integration and expectation by Fubini's theorem, which should be justified using a priori estimates on the stochastic quantities using the strong linear growth assumptions.